nLab
model structure on orthogonal spectra

Contents

Context

Model category theory

model category

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

Presentation of (,1)(\infty,1)-categories

Model structures

for \infty-groupoids

for ∞-groupoids

for nn-groupoids

for \infty-groups

for \infty-algebras

general

specific

for stable/spectrum objects

for (,1)(\infty,1)-categories

for stable (,1)(\infty,1)-categories

for (,1)(\infty,1)-operads

for (n,r)(n,r)-categories

for (,1)(\infty,1)-sheaves / \infty-stacks

Stable homotopy theory

Contents

Idea

A model structure on spectra for orthogonal spectra.

The category of orthogonal spectra is a presentation of the symmetric monoidal (∞,1)-category of spectra, with the special property that it implements the smash product of spectra such as to yield itself a symmetric monoidal model category of spectra: the model structure on orthogonal spectra. This implies in particular that with respect to this symmetric smash product of spectra an E-∞ ring is presented simply as a plain commutative monoid in orthogonal spectra.

Homotopy theory of orthogonal spectra

Categorical algebra

When defining a commutative ring as an abelian group AA equipped with an associative, commutative and untial bilinear pairing

A A()()A A \otimes_{\mathbb{Z}} A \overset{(-)\cdot (-)}{\longrightarrow} A

one evidently makes crucial use of the tensor product of abelian groups \otimes_{\mathbb{Z}}. That tensor product itself gives the category Ab of all abelian groups a structure similar to that of a ring, namely it equips it with a pairing

Ab×Ab() ()Ab Ab \times Ab \overset{(-)\otimes_{\mathbb{Z}}(-)}{\longrightarrow} Ab

that is a functor out of the product category of Ab with itself, satisfying category-theoretic analogs of the properties of associativity, commutativity and unitality.

One says that a ring AA is a commutative monoid in the category Ab of abelian groups, and that this concept makes sense since AbAb itself is a symmetric monoidal category.

Now in stable homotopy theory, as we have seen above, the category Ab is improved to the stable homotopy category Ho(Spectra)Ho(Spectra) (def. ), or rather to any stable model structure on spectra presenting it. Hence in order to correspondingly refine commutative monoids in Ab (namely commutative rings) to commutative monoids in Ho(Spectra) (namely commutative ring spectra), there needs to be a suitable symmetric monoidal category structure on the category of spectra. Its analog of the tensor product of abelian groups is to be called the symmetric monoidal smash product of spectra. The problem is how to construct it.

The theory for handling such a problem is categorical algebra. Here we discuss the minimum of categorical algebra that will allow us to elegantly construct the symmetric monoidal smash product of spectra.

Monoidal topological categories

We want to lift the concepts of ring and module from abelian groups to spectra. This requires a general idea of what it means to generalize these concepts at all. The abstract theory of such generalizations is that of monoid in a monoidal category.

We recall the basic definitions of monoidal categories and of monoids and modules internal to monoidal categories. We list archetypical examples at the end of this section, starting with example below. These examples are all fairly immediate. The point of the present discussion is to construct the non-trivial example of Day convolution monoidal stuctures below.

Definition

A (pointed) topologically enriched monoidal category is a (pointed) topologically enriched category 𝒞\mathcal{C} (def.) equipped with

  1. a (pointed) topologically enriched functor (def.)

    :𝒞×𝒞𝒞 \otimes \;\colon\; \mathcal{C} \times \mathcal{C} \longrightarrow \mathcal{C}

    out of the (pointed) topologival product category of 𝒞\mathcal{C} with itself (def. ), called the tensor product,

  2. an object

    1𝒞 1 \in \mathcal{C}

    called the unit object or tensor unit,

  3. a natural isomorphism (def.)

    a:(()())()()(()()) a \;\colon\; ((-)\otimes (-)) \otimes (-) \overset{\simeq}{\longrightarrow} (-) \otimes ((-)\otimes(-))

    called the associator,

  4. a natural isomorphism

    :(1())() \ell \;\colon\; (1 \otimes (-)) \overset{\simeq}{\longrightarrow} (-)

    called the left unitor, and a natural isomorphism

    r:()1() r \;\colon\; (-) \otimes 1 \overset{\simeq}{\longrightarrow} (-)

    called the right unitor,

such that the following two kinds of diagrams commute, for all objects involved:

  1. triangle identity:

    (x1)y a x,1,y x(1y) ρ x1 y 1 xλ y xy \array{ & (x \otimes 1) \otimes y &\stackrel{a_{x,1,y}}{\longrightarrow} & x \otimes (1 \otimes y) \\ & {}_{\rho_x \otimes 1_y}\searrow && \swarrow_{1_x \otimes \lambda_y} & \\ && x \otimes y && }
  2. the pentagon identity:

    (wx)(yz) α wx,y,z α w,x,yz ((wx)y)z (w(x(yz))) α w,x,yid z id wα x,y,z (w(xy))z α w,xy,z w((xy)z) \array{ && (w \otimes x) \otimes (y \otimes z) \\ & {}^{\mathllap{\alpha_{w \otimes x, y, z}}}\nearrow && \searrow^{\mathrlap{\alpha_{w,x,y \otimes z}}} \\ ((w \otimes x ) \otimes y) \otimes z && && (w \otimes (x \otimes (y \otimes z))) \\ {}^{\mathllap{\alpha_{w,x,y}} \otimes id_z }\downarrow && && \uparrow^{\mathrlap{ id_w \otimes \alpha_{x,y,z} }} \\ (w \otimes (x \otimes y)) \otimes z && \underset{\alpha_{w,x \otimes y, z}}{\longrightarrow} && w \otimes ( (x \otimes y) \otimes z ) }
Lemma

(Kelly 64)

Let (𝒞,,1)(\mathcal{C}, \otimes, 1) be a monoidal category, def. . Then the left and right unitors \ell and rr satisfy the following conditions:

  1. 1=r 1:111\ell_1 = r_1 \;\colon\; 1 \otimes 1 \overset{\simeq}{\longrightarrow} 1;

  2. for all objects x,y𝒞x,y \in \mathcal{C} the following diagrams commutes:

    (1x)y α 1,x,y xid y 1(xy) xy xy; \array{ (1 \otimes x) \otimes y & & \\ {}^\mathllap{\alpha_{1, x, y}} \downarrow & \searrow^\mathrlap{\ell_x \otimes id_y} & \\ 1 \otimes (x \otimes y) & \underset{\ell_{x \otimes y}}{\longrightarrow} & x \otimes y } \,;

    and

    x(y1) α 1,x,y 1 id xr y (xy)1 r xy xy; \array{ x \otimes (y \otimes 1) & & \\ {}^\mathllap{\alpha^{-1}_{1, x, y}} \downarrow & \searrow^\mathrlap{id_x \otimes r_y} & \\ (x \otimes y) \otimes 1 & \underset{r_{x \otimes y}}{\longrightarrow} & x \otimes y } \,;

For proof see at monoidal category this lemma and this lemma.

Remark

Just as for an associative algebra it is sufficient to demand 1a=a1 a = a and a1=aa 1 = a and (ab)c=a(bc)(a b) c = a (b c) in order to have that expressions of arbitrary length may be re-bracketed at will, so there is a coherence theorem for monoidal categories which states that all ways of composing the unitors and associators in a monoidal category (def. ) to go from one expression to another will coincide. Accordingly, much as one may drop the notation for the bracketing in an associative algebra altogether, so one may, with due care, reason about monoidal categories without always making all unitors and associators explicit.

Definition

A (pointed) topological braided monoidal category, is a (pointed) topological monoidal category 𝒞\mathcal{C} (def. ) equipped with a natural isomorphism

τ x,y:xyyx \tau_{x,y} \colon x \otimes y \to y \otimes x

called the braiding, such that the following two kinds of diagrams commute for all objects involved (“hexagon identities”):

(xy)z a x,y,z x(yz) τ x,yz (yz)x τ x,yId a y,z,x (yx)z a y,x,z y(xz) Idτ x,z y(zx) \array{ (x \otimes y) \otimes z &\stackrel{a_{x,y,z}}{\to}& x \otimes (y \otimes z) &\stackrel{\tau_{x,y \otimes z}}{\to}& (y \otimes z) \otimes x \\ \downarrow^{\tau_{x,y}\otimes Id} &&&& \downarrow^{a_{y,z,x}} \\ (y \otimes x) \otimes z &\stackrel{a_{y,x,z}}{\to}& y \otimes (x \otimes z) &\stackrel{Id \otimes \tau_{x,z}}{\to}& y \otimes (z \otimes x) }

and

x(yz) a x,y,z 1 (xy)z τ xy,z z(xy) Idτ y,z a z,x,y 1 x(zy) a x,z,y 1 (xz)y τ x,zId (zx)y, \array{ x \otimes (y \otimes z) &\stackrel{a^{-1}_{x,y,z}}{\to}& (x \otimes y) \otimes z &\stackrel{\tau_{x \otimes y, z}}{\to}& z \otimes (x \otimes y) \\ \downarrow^{Id \otimes \tau_{y,z}} &&&& \downarrow^{a^{-1}_{z,x,y}} \\ x \otimes (z \otimes y) &\stackrel{a^{-1}_{x,z,y}}{\to}& (x \otimes z) \otimes y &\stackrel{\tau_{x,z} \otimes Id}{\to}& (z \otimes x) \otimes y } \,,

where a x,y,z:(xy)zx(yz)a_{x,y,z} \colon (x \otimes y) \otimes z \to x \otimes (y \otimes z) denotes the components of the associator of 𝒞 \mathcal{C}^\otimes.

Definition

A (pointed) topological symmetric monoidal category is a (pointed) topological braided monoidal category (def. ) for which the braiding

τ x,y:xyyx \tau_{x,y} \colon x \otimes y \to y \otimes x

satisfies the condition:

τ y,xτ x,y=1 xy \tau_{y,x} \circ \tau_{x,y} = 1_{x \otimes y}

for all objects x,yx, y

Definition

Given a (pointed) topological symmetric monoidal category 𝒞\mathcal{C} with tensor product \otimes (def. ) it is called a closed monoidal category if for each Y𝒞Y \in \mathcal{C} the functor Y()()YY \otimes(-)\simeq (-)\otimes Y has a right adjoint, denoted hom(Y,)hom(Y,-)

𝒞hom(Y,)()Y𝒞, \mathcal{C} \underoverset {\underset{hom(Y,-)}{\longrightarrow}} {\overset{(-) \otimes Y}{\longleftarrow}} {\bot} \mathcal{C} \,,

hence if there are natural bijections

Hom 𝒞(XY,Z)Hom 𝒞C(X,hom(Y,Z)) Hom_{\mathcal{C}}(X \otimes Y, Z) \;\simeq\; Hom_{\mathcal{C}}{C}(X, hom(Y,Z))

for all objects X,Z𝒞X,Z \in \mathcal{C}.

Since for the case that X=1X = 1 is the tensor unit of 𝒞\mathcal{C} this means that

Hom 𝒞(1,hom(Y,Z))Hom 𝒞(Y,Z), Hom_{\mathcal{C}}(1,hom(Y,Z)) \simeq Hom_{\mathcal{C}}(Y,Z) \,,

the object hom(Y,Z)𝒞hom(Y,Z) \in \mathcal{C} is an enhancement of the ordinary hom-set Hom 𝒞(Y,Z)Hom_{\mathcal{C}}(Y,Z) to an object in 𝒞\mathcal{C}. Accordingly, it is also called the internal hom between YY and ZZ.

In a closed monoidal category, the adjunction isomorphism between tensor product and internal hom even holds internally:

Proposition

In a symmetric closed monoidal category (def. ) there are natural isomorphisms

hom(XY,Z)hom(X,hom(Y,Z)) hom(X \otimes Y, Z) \;\simeq\; hom(X, hom(Y,Z))

whose image under Hom 𝒞(1,)Hom_{\mathcal{C}}(1,-) are the defining natural bijections of def. .

Proof

Let A𝒞A \in \mathcal{C} be any object. By applying the defining natural bijections twice, there are composite natural bijections

Hom 𝒞(A,hom(XY,Z)) Hom 𝒞(A(XY),Z) Hom 𝒞((AX)Y,Z) Hom 𝒞(AX,hom(Y,Z)) Hom 𝒞(A,hom(X,hom(Y,Z))). \begin{aligned} Hom_{\mathcal{C}}(A , hom(X \otimes Y, Z)) & \simeq Hom_{\mathcal{C}}(A \otimes (X \otimes Y), Z) \\ & \simeq Hom_{\mathcal{C}}((A \otimes X)\otimes Y, Z) \\ & \simeq Hom_{\mathcal{C}}(A \otimes X, hom(Y,Z)) \\ & \simeq Hom_{\mathcal{C}}(A, hom(X,hom(Y,Z))) \end{aligned} \,.

Since this holds for all AA, the Yoneda lemma (the fully faithfulness of the Yoneda embedding) says that there is an isomorphism hom(XY,Z)hom(X,hom(Y,Z))hom(X\otimes Y, Z) \simeq hom(X,hom(Y,Z)). Moreover, by taking A=1A = 1 in the above and using the left unitor isomorphisms A(XY)XYA \otimes (X \otimes Y) \simeq X \otimes Y and AXXA\otimes X \simeq X we get a commuting diagram

Hom 𝒞(1,hom(XY,)) Hom 𝒞(1,hom(X,hom(Y,Z))) Hom 𝒞(XY,Z) Hom 𝒞(X,hom(Y,Z)). \array{ Hom_{\mathcal{C}}(1,hom(X\otimes Y, )) &\overset{\simeq}{\longrightarrow}& Hom_{\mathcal{C}}(1,hom(X,hom(Y,Z))) \\ {}^{\mathllap{\simeq}}\downarrow && \downarrow^{\mathrlap{\simeq}} \\ Hom_{\mathcal{C}}(X \otimes Y, Z) &\overset{\simeq}{\longrightarrow}& Hom_{\mathcal{C}}(X, hom(Y,Z)) } \,.
Example

The category Set of sets and functions between them, regarded as enriched in discrete topological spaces, becomes a symmetric monoidal category according to def. with tensor product the Cartesian product ×\times of sets. The associator, unitor and braiding isomorphism are the evident (almost unnoticable but nevertheless nontrivial) canonical identifications.

Similarly the category Top cgTop_{cg} of compactly generated topological spaces (def.) becomes a symmetric monoidal category with tensor product the corresponding Cartesian products, hence the operation of forming k-ified (cor.) product topological spaces (exmpl.). The underlying functions of the associator, unitor and braiding isomorphisms are just those of the underlying sets, as above.

Symmetric monoidal categories, such as these, for which the tensor product is the Cartesian product are called Cartesian monoidal categories.

Both examples are closed monoidal categories (def. ), with internal hom the mapping spaces (prop.).

Example

The category Top cg */Top_{cg}^{\ast/} of pointed compactly generated topological spaces with tensor product the smash product \wedge (def.)

XYX×YXY X \wedge Y \coloneqq \frac{X\times Y}{X\vee Y}

is a symmetric monoidal category (def. ) with unit object the pointed 0-sphere S 0S^0.

The components of the associator, the unitors and the braiding are those of Top as in example , descended to the quotient topological spaces which appear in the definition of the smash product. This works for pointed compactly generated spaces (but not for general pointed topological spaces) by this prop..

The category Top cg */Top^{\ast/}_{cg} is also a closed monoidal category (def. ), with internal hom the pointed mapping space Maps(,) *Maps(-,-)_\ast (exmpl.)

Example

The category Ab of abelian groups, regarded as enriched in discrete topological spaces, becomes a symmetric monoidal category with tensor product the actual tensor product of abelian groups \otimes_{\mathbb{Z}} and with tensor unit the additive group \mathbb{Z} of integers. Again the associator, unitor and braiding isomorphism are the evident ones coming from the underlying sets, as in example .

This is a closd monoidal cagory? with internal hom hom(A,B)hom(A,B) being the set of homomorphisms Hom Ab(A,B)Hom_{Ab}(A,B) equipped with the pointwise group structure for ϕ 1,ϕ 2Hom Ab(A,B)\phi_1, \phi_2 \in Hom_{Ab}(A,B) then (ϕ 1+ϕ 2)(a)ϕ 1(a)+ϕ 2(b)B(\phi_1 + \phi_2)(a) \coloneqq \phi_1(a) + \phi_2(b) \; \in B.

This is the archetypical case that motivates the notation “\otimes” for the pairing operation in a monoidal category:

Example

The category category of chain complexes Ch Ch_\bullet, equipped with the tensor product of chain complexes is a symmetric monoidal category (def. ).

In this case the braiding has a genuinely non-trivial aspect to it, beyond just the swapping of coordinates as in examples , and def. , namely for X,YCh X, Y \in Ch_\bullet then

(XY) n=n 1+n 2=nX n 1 X n 2 (X \otimes Y)_n = \underset{n_1 + n_2 = n}{\otimes} X_{n_1} \otimes_{\mathbb{Z}} X_{n_2}

and in these components the braiding isomorphism is that of Ab, but with a minus sign thrown in whener two odd-graded components are commuted.

This is a first shadow of the graded-commutativity that also exhibited by spectra.

(e.g. Hovey 99, prop. 4.2.13)

Algebras and modules
Definition

Given a (pointed) topological monoidal category (𝒞,,1)(\mathcal{C}, \otimes, 1), then a monoid internal to (𝒞,,1)(\mathcal{C}, \otimes, 1) is

  1. an object A𝒞A \in \mathcal{C};

  2. a morphism e:1Ae \;\colon\; 1 \longrightarrow A (called the unit)

  3. a morphism μ:AAA\mu \;\colon\; A \otimes A \longrightarrow A (called the product);

such that

  1. (associativity) the following diagram commutes

    (AA)A a A,A,A A(AA) Aμ AA μA μ AA μ A, \array{ (A\otimes A) \otimes A &\underoverset{\simeq}{a_{A,A,A}}{\longrightarrow}& A \otimes (A \otimes A) &\overset{A \otimes \mu}{\longrightarrow}& A \otimes A \\ {}^{\mathllap{\mu \otimes A}}\downarrow && && \downarrow^{\mathrlap{\mu}} \\ A \otimes A &\longrightarrow& &\overset{\mu}{\longrightarrow}& A } \,,

    where aa is the associator isomorphism of 𝒞\mathcal{C};

  2. (unitality) the following diagram commutes:

    1A eid AA ide A1 μ r A, \array{ 1 \otimes A &\overset{e \otimes id}{\longrightarrow}& A \otimes A &\overset{id \otimes e}{\longleftarrow}& A \otimes 1 \\ & {}_{\mathllap{\ell}}\searrow & \downarrow^{\mathrlap{\mu}} & & \swarrow_{\mathrlap{r}} \\ && A } \,,

    where \ell and rr are the left and right unitor isomorphisms of 𝒞\mathcal{C}.

Moreover, if (𝒞,,1)(\mathcal{C}, \otimes , 1) has the structure of a symmetric monoidal category (def. ) (𝒞,,1,B)(\mathcal{C}, \otimes, 1, B) with symmetric braiding τ\tau, then a monoid (A,μ,e)(A,\mu, e) as above is called a commutative monoid in (𝒞,,1,B)(\mathcal{C}, \otimes, 1, B) if in addition

  • (commutativity) the following diagram commutes

    AA τ A,A AA μ μ A. \array{ A \otimes A && \underoverset{\simeq}{\tau_{A,A}}{\longrightarrow} && A \otimes A \\ & {}_{\mathllap{\mu}}\searrow && \swarrow_{\mathrlap{\mu}} \\ && A } \,.

A homomorphism of monoids (A 1,μ 1,e 1)(A 2,μ 2,f 2)(A_1, \mu_1, e_1)\longrightarrow (A_2, \mu_2, f_2) is a morphism

f:A 1A 2 f \;\colon\; A_1 \longrightarrow A_2

in 𝒞\mathcal{C}, such that the following two diagrams commute

A 1A 1 ff A 2A 2 μ 1 μ 2 A 1 f A 2 \array{ A_1 \otimes A_1 &\overset{f \otimes f}{\longrightarrow}& A_2 \otimes A_2 \\ {}^{\mathllap{\mu_1}}\downarrow && \downarrow^{\mathrlap{\mu_2}} \\ A_1 &\underset{f}{\longrightarrow}& A_2 }

and

1 𝒸 e 1 A 1 e 2 f A 2. \array{ 1_{\mathcal{c}} &\overset{e_1}{\longrightarrow}& A_1 \\ & {}_{\mathllap{e_2}}\searrow & \downarrow^{\mathrlap{f}} \\ && A_2 } \,.

Write Mon(𝒞,,1)Mon(\mathcal{C}, \otimes,1) for the category of monoids in 𝒞\mathcal{C}, and CMon(𝒞,,1)CMon(\mathcal{C}, \otimes, 1) for its subcategory of commutative monoids.

Example

Given a (pointed) topological monoidal category (𝒞,,1)(\mathcal{C}, \otimes, 1), then the tensor unit 11 is a monoid in 𝒞\mathcal{C} (def. ) with product given by either the left or right unitor

1=r 1:111. \ell_1 = r_1 \;\colon\; 1 \otimes 1 \overset{\simeq}{\longrightarrow} 1 \,.

By lemma , these two morphisms coincide and define an associative product with unit the identity id:11id \colon 1 \to 1.

If (𝒞,,1)(\mathcal{C}, \otimes , 1) is a symmetric monoidal category (def. ), then this monoid is a commutative monoid.

Example

Given a symmetric monoidal category (𝒞,,1)(\mathcal{C}, \otimes, 1) (def. ), and given two commutative monoids (E i,μ i,e i)(E_i, \mu_i, e_i) i{1,2}i \in \{1,2\} (def. {MonoidsInMonoidalCategory}), then the tensor product E 1E 2E_1 \otimes E_2 becomes itself a commutative monoid with unit morphism

e:111e 1e 2E 1E 2 e \;\colon\; 1 \overset{\simeq}{\longrightarrow} 1 \otimes 1 \overset{e_1 \otimes e_2}{\longrightarrow} E_1 \otimes E_2

(where the first isomorphism is, 1 1=r 1 1\ell_1^{-1} = r_1^{-1} (lemma )) and with product morphism given by

E 1E 2E 1E 2idτ E 2,E 1idE 1E 1E 2E 2μ 1μ 2E 1E 2 E_1 \otimes E_2 \otimes E_1 \otimes E_2 \overset{id \otimes \tau_{E_2, E_1} \otimes id}{\longrightarrow} E_1 \otimes E_1 \otimes E_2 \otimes E_2 \overset{\mu_1 \otimes \mu_2}{\longrightarrow} E_1 \otimes E_2

(where we are notationally suppressing the associators and where τ\tau denotes the braiding of $\mathcal{C} ).

That this definition indeed satisfies associativity and commutativity follows from the corresponding properties of (E i,μ i,e i)(E_i,\mu_i, e_i), and from the hexagon identities for the braiding (def. ) and from symmetry of the braiding.

Definition

Given a (pointed) [[topologically enriched category|topological]] [[monoidal category]] (𝒞,,1)(\mathcal{C}, \otimes, 1) (def. ), and given (A,μ,e)(A,\mu,e) a [[monoid in a monoidal category|monoid in]] (𝒞,,1)(\mathcal{C}, \otimes, 1) (def. ), then a left [[module object]] in (𝒞,,1)(\mathcal{C}, \otimes, 1) over (A,μ,e)(A,\mu,e) is

  1. an [[object]] N𝒞N \in \mathcal{C};

  2. a [[morphism]] ρ:ANN\rho \;\colon\; A \otimes N \longrightarrow N (called the [[action]]);

such that

  1. ([[unitality]]) the following [[commuting diagram|diagram commutes]]:

    1N eid AN ρ N, \array{ 1 \otimes N &\overset{e \otimes id}{\longrightarrow}& A \otimes N \\ & {}_{\mathllap{\ell}}\searrow & \downarrow^{\mathrlap{\rho}} \\ && N } \,,

    where \ell is the left unitor isomorphism of 𝒞\mathcal{C}.

  2. (action property) the following [[commuting diagram|diagram commutes]]

    (AA)N a A,A,N A(AN) Aρ AN μN ρ AN ρ N, \array{ (A\otimes A) \otimes N &\underoverset{\simeq}{a_{A,A,N}}{\longrightarrow}& A \otimes (A \otimes N) &\overset{A \otimes \rho}{\longrightarrow}& A \otimes N \\ {}^{\mathllap{\mu \otimes N}}\downarrow && && \downarrow^{\mathrlap{\rho}} \\ A \otimes N &\longrightarrow& &\overset{\rho}{\longrightarrow}& N } \,,

A [[homomorphism]] of left AA-module objects

(N 1,ρ 1)(N 2,ρ 2) (N_1, \rho_1) \longrightarrow (N_2, \rho_2)

is a morphism

f:N 1N 2 f\;\colon\; N_1 \longrightarrow N_2

in 𝒞\mathcal{C}, such that the following [[commuting diagram|diagram commutes]]:

AN 1 Af AN 2 ρ 1 ρ 2 N 1 f N 2. \array{ A\otimes N_1 &\overset{A \otimes f}{\longrightarrow}& A\otimes N_2 \\ {}^{\mathllap{\rho_1}}\downarrow && \downarrow^{\mathrlap{\rho_2}} \\ N_1 &\underset{f}{\longrightarrow}& N_2 } \,.

For the resulting [[category of modules]] of left AA-modules in 𝒞\mathcal{C} with AA-module homomorphisms between them, we write

AMod(𝒞). A Mod(\mathcal{C}) \,.

This is naturally a (pointed) [[topologically enriched category]] itself.

Example

Given a [[monoidal category]] (𝒞,,1)(\mathcal{C},\otimes, 1) (def. ) with the [[tensor unit]] 11 regarded as a [[monoid in a monoidal category]] via example , then the left [[unitor]]

C:1CC \ell_C \;\colon\; 1\otimes C \longrightarrow C

makes every object C𝒞C \in \mathcal{C} into a left module, according to def. , over CC. The action property holds due to lemma . This gives an [[equivalence of categories]]

𝒞1Mod(𝒞) \mathcal{C} \simeq 1 Mod(\mathcal{C})

of 𝒞\mathcal{C} with the [[category of modules]] over its tensor unit.

Example

The archetypical case in which all these abstract concepts reduce to the basic familiar ones is the symmetric monoidal category [[Ab]] of [[abelian groups]] from example .

  1. A [[monoid in a monoidal category|monoid in]] (Ab, ,)(Ab, \otimes_{\mathbb{Z}}, \mathbb{Z}) (def. ) is equivalently a [[ring]].

  2. A [[commutative monoid in a symmetric monoidal category|commutative monoid in]] in (Ab, ,)(Ab, \otimes_{\mathbb{Z}}, \mathbb{Z}) (def. ) is equivalently a [[commutative ring]] RR.

  3. An RR-[[module object]] in (Ab, ,)(Ab, \otimes_{\mathbb{Z}}, \mathbb{Z}) (def. ) is equivalently an RR-[[module]];

  4. The tensor product of RR-module objects (def. ) is the standard [[tensor product of modules]].

  5. The [[category of modules|category of module objects]] RMod(Ab)R Mod(Ab) (def. ) is the standard [[category of modules]] RModR Mod.

Example

Closely related to the example , but closer to the structure we will see below for spectra, are [[monoid in a monoidal category|monoids]] in the [[category of chain complexes]] (Ch ,,)(Ch_\bullet, \otimes, \mathbb{Z}) from example . These monoids are equivalently [[differential graded algebras]].

Proposition

In the situation of def. , the monoid (A,μ,e)(A,\mu, e) canonically becomes a left module over itself by setting ρμ\rho \coloneqq \mu. More generally, for C𝒞C \in \mathcal{C} any object, then ACA \otimes C naturally becomes a left AA-module by setting:

ρ:A(AC)a A,A,C 1(AA)CμidAC. \rho \;\colon\; A \otimes (A \otimes C) \underoverset{\simeq}{a^{-1}_{A,A,C}}{\longrightarrow} (A \otimes A) \otimes C \overset{\mu \otimes id}{\longrightarrow} A \otimes C \,.

The AA-modules of this form are called [[free modules]].

The [[free functor]] FF constructing free AA-modules is [[left adjoint]] to the [[forgetful functor]] UU which sends a module (N,ρ)(N,\rho) to the underlying object U(N,ρ)NU(N,\rho) \coloneqq N.

AMod(𝒞)UF𝒞. A Mod(\mathcal{C}) \underoverset {\underset{U}{\longrightarrow}} {\overset{F}{\longleftarrow}} {\bot} \mathcal{C} \,.
Proof

A homomorphism out of a free AA-module is a morphism in 𝒞\mathcal{C} of the form

f:ACN f \;\colon\; A\otimes C \longrightarrow N

fitting into the diagram (where we are notationally suppressing the [[associator]])

AAC Af AN μid ρ AC f N. \array{ A \otimes A \otimes C &\overset{A \otimes f}{\longrightarrow}& A \otimes N \\ {}^{\mathllap{\mu \otimes id}}\downarrow && \downarrow^{\mathrlap{\rho}} \\ A \otimes C &\underset{f}{\longrightarrow}& N } \,.

Consider the composite

f˜:C C1CeidACfN, \tilde f \;\colon\; C \underoverset{\simeq}{\ell_C}{\longrightarrow} 1 \otimes C \overset{e\otimes id}{\longrightarrow} A \otimes C \overset{f}{\longrightarrow} N \,,

i.e. the restriction of ff to the unit “in” AA. By definition, this fits into a [[commuting square]] of the form (where we are now notationally suppressing the [[associator]] and the [[unitor]])

AC idf˜ AN ideid = AAC idf AN. \array{ A \otimes C &\overset{id \otimes \tilde f}{\longrightarrow}& A \otimes N \\ {}^{\mathllap{id \otimes e \otimes id}}\downarrow && \downarrow^{\mathrlap{=}} \\ A \otimes A \otimes C &\underset{id \otimes f}{\longrightarrow}& A \otimes N } \,.

Pasting this square onto the top of the previous one yields

AC idf˜ AN ideid = AAC Af AN μid ρ AC f N, \array{ A \otimes C &\overset{id \otimes \tilde f}{\longrightarrow}& A \otimes N \\ {}^{\mathllap{id \otimes e \otimes id}}\downarrow && \downarrow^{\mathrlap{=}} \\ A \otimes A \otimes C &\overset{A \otimes f}{\longrightarrow}& A \otimes N \\ {}^{\mathllap{\mu \otimes id}}\downarrow && \downarrow^{\mathrlap{\rho}} \\ A \otimes C &\underset{f}{\longrightarrow}& N } \,,

where now the left vertical composite is the identity, by the unit law in AA. This shows that ff is uniquely determined by f˜\tilde f via the relation

f=ρ(id Af˜). f = \rho \circ (id_A \otimes \tilde f) \,.

This natural bijection between ff and f˜\tilde f establishes the adjunction.

Definition

Given a (pointed) [[topologically enriched category|topological]] [[closed monoidal category|closed]] [[symmetric monoidal category]] (𝒞,,1)(\mathcal{C}, \otimes, 1) (def. , def. ), given (A,μ,e)(A,\mu,e) a [[commutative monoid in a symmetric monoidal category|commutative monoid in]] (𝒞,,1)(\mathcal{C}, \otimes, 1) (def. ), and given (N 1,ρ 1)(N_1, \rho_1) and (N 2,ρ 2)(N_2, \rho_2) two left AA-[[module objects]] (def.), then

  1. the [[tensor product of modules]] N 1 AN 2N_1 \otimes_A N_2 is, if it exists, the [[coequalizer]]

    N 1AN 2AAAAρ 1(τ N 1,AN 2)N 1ρ 2N 1N 1coeqN 1 AN 2 N_1 \otimes A \otimes N_2 \underoverset {\underset{\rho_{1}\circ (\tau_{N_1,A} \otimes N_2)}{\longrightarrow}} {\overset{N_1 \otimes \rho_2}{\longrightarrow}} {\phantom{AAAA}} N_1 \otimes N_1 \overset{coeq}{\longrightarrow} N_1 \otimes_A N_2

    and if A()A \otimes (-) preserves these coequalizers, then this is equipped with the left AA-action induced from the left AA-action on N 1N_1

  2. the function module hom A(N 1,N 2)hom_A(N_1,N_2) is, if it exists, the [[equalizer]]

    hom A(N 1,N 2)equhom(N 1,N 2)AAAAAAhom(AN 1,ρ 2)(A())hom(ρ 1,N 2)hom(AN 1,N 2). hom_A(N_1, N_2) \overset{equ}{\longrightarrow} hom(N_1, N_2) \underoverset {\underset{hom(A \otimes N_1, \rho_2)\circ (A \otimes(-))}{\longrightarrow}} {\overset{hom(\rho_1,N_2)}{\longrightarrow}} {\phantom{AAAAAA}} hom(A \otimes N_1, N_2) \,.

    equipped with the left AA-action that is induced by the left AA-action on N 2N_2 via

    Ahom(X,N 2)hom(X,N 2)Ahom(X,N 2)XidevAN 2ρ 2N 2. \frac{ A \otimes hom(X,N_2) \longrightarrow hom(X,N_2) }{ A \otimes hom(X,N_2) \otimes X \overset{id \otimes ev}{\longrightarrow} A \otimes N_2 \overset{\rho_2}{\longrightarrow} N_2 } \,.

(e.g. Hovey-Shipley-Smith 00, lemma 2.2.2 and lemma 2.2.8)

Proposition

Given a (pointed) [[topologically enriched category|topological]] [[closed monoidal category|closed]] [[symmetric monoidal category]] (𝒞,,1)(\mathcal{C}, \otimes, 1) (def. , def. ), and given (A,μ,e)(A,\mu,e) a [[commutative monoid in a symmetric monoidal category|commutative monoid in]] (𝒞,,1)(\mathcal{C}, \otimes, 1) (def. ). If all [[coequalizers]] exist in 𝒞\mathcal{C}, then the [[tensor product of modules]] A\otimes_A from def. makes the [[category of modules]] AMod(𝒞)A Mod(\mathcal{C}) into a [[symmetric monoidal category]], (AMod, A,A)(A Mod, \otimes_A, A) with [[tensor unit]] the object AA itself, regarded as an AA-module via prop. .

If moreover all [[equalizers]] exist, then this is a [[closed monoidal category]] (def. ) with [[internal hom]] given by the function modules hom Ahom_A of def. .

(e.g. Hovey-Shipley-Smith 00, lemma 2.2.2, lemma 2.2.8)

Proof sketch

The associators and braiding for A\otimes_{A} are induced directly from those of \otimes and the [[universal property]] of [[coequalizers]]. That AA is the tensor unit for A\otimes_{A} follows with the same kind of argument that we give in the proof of example below.

Example

For (A,μ,e)(A,\mu,e) a [[monoid in a monoidal category|monoid]] (def. ) in a [[symmetric monoidal category]] (𝒞,,1)(\mathcal{C},\otimes, 1) (def. ), the [[tensor product of modules]] (def. ) of two [[free modules]] (def. ) AC 1A\otimes C_1 and AC 2A \otimes C_2 always exists and is the free module over the tensor product in 𝒞\mathcal{C} of the two generators:

(AC 1) A(AC 2)A(C 1C 2). (A \otimes C_1) \otimes_A (A \otimes C_2) \simeq A \otimes (C_1 \otimes C_2) \,.

Hence if 𝒞\mathcal{C} has all [[coequalizers]], so that the [[category of modules]] is a [[monoidal category]] (AMod, A,A)(A Mod, \otimes_A, A) (prop. ) then the free module functor (def. ) is a [[strong monoidal functor]] (def. )

F:(𝒞,,1)(AMod, A,A). F \;\colon\; (\mathcal{C}, \otimes, 1) \longrightarrow (A Mod, \otimes_A, A) \,.
Proof

It is sufficient to show that the diagram

AAAAAAAidμμidAAμA A \otimes A \otimes A \underoverset {\underset{id \otimes \mu}{\longrightarrow}} {\overset{\mu \otimes id}{\longrightarrow}} {\phantom{AAAA}} A \otimes A \overset{\mu}{\longrightarrow} A

is a [[coequalizer]] diagram (we are notationally suppressing the [[associators]]), hence that A AAAA \otimes_A A \simeq A, hence that the claim holds for C 1=1C_1 = 1 and C 2=1C_2 = 1.

To that end, we check the [[universal property]] of the [[coequalizer]]:

First observe that μ\mu indeed coequalizes idμid \otimes \mu with μid\mu \otimes id, since this is just the [[associativity]] clause in def. . So for f:AAQf \colon A \otimes A \longrightarrow Q any other morphism with this property, we need to show that there is a unique morphism ϕ:AQ\phi \colon A \longrightarrow Q which makes this [[commuting diagram|diagram commute]]:

AA μ A f ϕ Q. \array{ A \otimes A &\overset{\mu}{\longrightarrow}& A \\ {}^{\mathllap{f}}\downarrow & \swarrow_{\mathrlap{\phi}} \\ Q } \,.

We claim that

ϕ:Ar 1A1ideAAfQ, \phi \;\colon\; A \underoverset{\simeq}{r^{-1}}{\longrightarrow} A \otimes 1 \overset{id \otimes e}{\longrightarrow} A \otimes A \overset{f}{\longrightarrow} Q \,,

where the first morphism is the inverse of the right [[unitor]] of 𝒞\mathcal{C}.

First to see that this does make the required triangle commute, consider the following pasting composite of [[commuting diagrams]]

AA μ A idr 1 r 1 AA1 μid A1 ide ide AAA μid AA idμ f AA f Q. \array{ A \otimes A &\overset{\mu}{\longrightarrow}& A \\ {}^{\mathllap{id \otimes r^{-1}}}_{\mathllap{\simeq}}\downarrow && \downarrow^{\mathrlap{r^{-1}}}_{\simeq} \\ A \otimes A \otimes 1 &\overset{\mu \otimes id}{\longrightarrow}& A \otimes 1 \\ {}^{\mathllap{id \otimes e}}\downarrow && \downarrow^{\mathrlap{id \otimes e} } \\ A \otimes A \otimes A &\overset{\mu \otimes id}{\longrightarrow}& A \otimes A \\ {}^{\mathllap{id \otimes \mu}}\downarrow && \downarrow^{\mathrlap{f}} \\ A \otimes A &\underset{f}{\longrightarrow}& Q } \,.

Here the the top square is the [[natural transformation|naturality]] of the right [[unitor]], the middle square commutes by the functoriality of the tensor product :𝒞×𝒞𝒞\otimes \;\colon\; \mathcal{C}\times \mathcal{C} \longrightarrow \mathcal{C} and the definition of the [[product category]] (def. ), while the commutativity of the bottom square is the assumption that ff coequalizes idμid \otimes \mu with μid\mu \otimes id.

Here the right vertical composite is ϕ\phi, while, by unitality of (A,μ,e)(A,\mu ,e), the left vertical composite is the identity on AA, Hence the diagram says that ϕμ=f\phi \circ \mu = f, which we needed to show.

It remains to see that ϕ\phi is the unique morphism with this property for given ff. For that let q:AQq \colon A \to Q be any other morphism with qμ=f q\circ \mu = f. Then consider the [[commuting diagram]]

A1 A ide = AA μ A f q Q, \array{ A \otimes 1 &\overset{\simeq}{\longleftarrow}& A \\ {}^{\mathllap{id\otimes e}}\downarrow & \searrow^{\simeq} & \downarrow^{\mathrlap{=}} \\ A \otimes A &\overset{\mu}{\longrightarrow}& A \\ {}^{\mathllap{f}}\downarrow & \swarrow_{\mathrlap{q}} \\ Q } \,,

where the top left triangle is the unitality condition and the two isomorphisms are the right [[unitor]] and its inverse. The commutativity of this diagram says that q=ϕq = \phi.

Definition

Given a [[monoidal category|monoidal]] [[category of modules]] (AMod, A,A)(A Mod , \otimes_A , A) as in prop. , then a [[monoid in a monoidal category|monoid]] (E,μ,e)(E, \mu, e) in (AMod, A,A)(A Mod , \otimes_A , A) (def. ) is called an AA-[[associative algebra|algebra]].

Propposition

Given a [[monoidal category|monoidal]] [[category of modules]] (AMod, A,A)(A Mod , \otimes_A , A) in a [[monoidal category]] (𝒞,,1)(\mathcal{C},\otimes, 1) as in prop. , and an AA-algebra (E,μ,e)(E,\mu,e) (def. ), then there is an [[equivalence of categories]]

AAlg comm(𝒞)CMon(AMod)CMon(𝒞) A/ A Alg_{comm}(\mathcal{C}) \coloneqq CMon(A Mod) \simeq CMon(\mathcal{C})^{A/}

between the [[category of commutative monoids]] in AModA Mod and the [[coslice category]] of commutative monoids in 𝒞\mathcal{C} under AA, hence between commutative AA-algebras in 𝒞\mathcal{C} and commutative monoids EE in 𝒞\mathcal{C} that are equipped with a homomorphism of monoids AEA \longrightarrow E.

(e.g. EKMM 97, VII lemma 1.3)

Proof

In one direction, consider a AA-algebra EE with unit e E:AEe_E \;\colon\; A \longrightarrow E and product μ E/A:E AEE\mu_{E/A} \colon E \otimes_A E \longrightarrow E. There is the underlying product μ E\mu_E

EAE AAA EE coeq E AE μ E μ E/A E. \array{ E \otimes A \otimes E & \underoverset {\underset{}{\longrightarrow}} {\overset{}{\longrightarrow}} {\phantom{AAA}} & E \otimes E &\overset{coeq}{\longrightarrow}& E \otimes_A E \\ && & {}_{\mathllap{\mu_E}}\searrow & \downarrow^{\mathrlap{\mu_{E/A}}} \\ && && E } \,.

By considering a diagram of such coequalizer diagrams with middle vertical morphism e Ee Ae_E\circ e_A, one find that this is a unit for μ E\mu_E and that (E,μ E,e Ee A)(E, \mu_E, e_E \circ e_A) is a commutative monoid in (𝒞,,1)(\mathcal{C}, \otimes, 1).

Then consider the two conditions on the unit e E:AEe_E \colon A \longrightarrow E. First of all this is an AA-module homomorphism, which means that

()AA ide E AE μ A ρ A e E E (\star) \;\;\;\;\; \;\;\;\;\; \array{ A \otimes A &\overset{id \otimes e_E}{\longrightarrow}& A \otimes E \\ {}^{\mathllap{\mu_A}}\downarrow && \downarrow^{\mathrlap{\rho}} \\ A &\underset{e_E}{\longrightarrow}& E }

[[commuting diagram|commutes]]. Moreover it satisfies the unit property

A AE e Aid E AE μ E/A E. \array{ A \otimes_A E &\overset{e_A \otimes id}{\longrightarrow}& E \otimes_A E \\ & {}_{\mathllap{\simeq}}\searrow & \downarrow^{\mathrlap{\mu_{E/A}}} \\ && E } \,.

By forgetting the tensor product over AA, the latter gives

AE eid EE A AE e Eid E AE μ E/A E = EAE e Eid EE ρ μ E E id E, \array{ A \otimes E &\overset{e \otimes id}{\longrightarrow}& E \otimes E \\ \downarrow && \downarrow^{\mathrlap{}} \\ A \otimes_A E &\overset{e_E \otimes id}{\longrightarrow}& E \otimes_A E \\ {}^{\mathllap{\simeq}}\downarrow && \downarrow^{\mathrlap{\mu_{E/A}}} \\ E &=& E } \;\;\;\;\;\;\;\; \simeq \;\;\;\;\;\;\;\; \array{ A \otimes E &\overset{e_E \otimes id}{\longrightarrow}& E \otimes E \\ {}^{\mathllap{\rho}}\downarrow && \downarrow^{\mathrlap{\mu_{E}}} \\ E &\underset{id}{\longrightarrow}& E } \,,

where the top vertical morphisms on the left the canonical coequalizers, which identifies the vertical composites on the right as shown. Hence this may be [[pasting|pasted]] to the square ()(\star) above, to yield a [[commuting square]]

AA ide E AE e Eid EE μ A ρ μ E A e E E id E=AA e Ee E EE μ A μ E A e E E. \array{ A \otimes A &\overset{id\otimes e_E}{\longrightarrow}& A \otimes E &\overset{e_E \otimes id}{\longrightarrow}& E \otimes E \\ {}^{\mathllap{\mu_A}}\downarrow && {}^{\mathllap{\rho}}\downarrow && \downarrow^{\mathrlap{\mu_{E}}} \\ A &\underset{e_E}{\longrightarrow}& E &\underset{id}{\longrightarrow}& E } \;\;\;\;\;\;\;\;\;\; = \;\;\;\;\;\;\;\;\;\; \array{ A \otimes A &\overset{e_E \otimes e_E}{\longrightarrow}& E \otimes E \\ {}^{\mathllap{\mu_A}}\downarrow && \downarrow^{\mathrlap{\mu_E}} \\ A &\underset{e_E}{\longrightarrow}& E } \,.

This shows that the unit e Ae_A is a homomorphism of monoids (A,μ A,e A)(E,μ E,e Ee A)(A,\mu_A, e_A) \longrightarrow (E, \mu_E, e_E\circ e_A).

Now for the converse direction, assume that (A,μ A,e A)(A,\mu_A, e_A) and (E,μ E,e E)(E, \mu_E, e'_E) are two commutative monoids in (𝒞,,1)(\mathcal{C}, \otimes, 1) with e E:AEe_E \;\colon\; A \to E a monoid homomorphism. Then EE inherits a left AA-[[module]] structure by

ρ:AEe AidEEμ EE. \rho \;\colon\; A \otimes E \overset{e_A \otimes id}{\longrightarrow} E \otimes E \overset{\mu_E}{\longrightarrow} E \,.

By commutativity and associativity it follows that μ E\mu_E coequalizes the two induced morphisms EAEAAEEE \otimes A \otimes E \underoverset{\longrightarrow}{\longrightarrow}{\phantom{AA}} E \otimes E. Hence the [[universal property]] of the [[coequalizer]] gives a factorization through some μ E/A:E AEE\mu_{E/A}\colon E \otimes_A E \longrightarrow E. This shows that (E,μ E/A,e E)(E, \mu_{E/A}, e_E) is a commutative AA-algebra.

Finally one checks that these two constructions are inverses to each other, up to isomorphism.

Topological ends and coends

For working with pointed [[topologically enriched functors]], a certain shape of [[limits]]/[[colimits]] is particularly relevant: these are called (pointed topological enriched) [[ends]] and [[coends]]. We here introduce these and then derive some of their basic properties, such as notably the expression for topological [[left Kan extension]] in terms of [[coends]] (prop. below). Further below it is via left Kan extension along the ordinary smash product of pointed topological spaces (“[[Day convolution]]”) that the [[symmetric monoidal smash product of spectra]] is induced.

Definition

Let 𝒞,𝒟\mathcal{C}, \mathcal{D} be pointed [[topologically enriched categories]] (def.), i.e. [[enriched categories]] over (Top cg */,,S 0)(Top_{cg}^{\ast/}, \wedge, S^0) from example .

  1. The pointed topologically enriched [[opposite category]] 𝒞 op\mathcal{C}^{op} is the [[topologically enriched category]] with the same [[objects]] as 𝒞\mathcal{C}, with [[hom-spaces]]

    𝒞 op(X,Y)𝒞(Y,X) \mathcal{C}^{op}(X,Y) \coloneqq \mathcal{C}(Y,X)

    and with [[composition]] given by [[braiding]] followed by the composition in 𝒞\mathcal{C}:

    𝒞 op(X,Y)𝒞 op(Y,Z)=𝒞(Y,X)𝒞(Z,Y)τ𝒞(Z,Y)𝒞(Y,X) Z,Y,X𝒞(Z,X)=𝒞 op(X,Z). \mathcal{C}^{op}(X,Y) \wedge \mathcal{C}^{op}(Y,Z) = \mathcal{C}(Y,X)\wedge \mathcal{C}(Z,Y) \underoverset{\simeq}{\tau}{\longrightarrow} \mathcal{C}(Z,Y) \wedge \mathcal{C}(Y,X) \overset{\circ_{Z,Y,X}}{\longrightarrow} \mathcal{C}(Z,X) = \mathcal{C}^{op}(X,Z) \,.
  2. the pointed topological [[product category]] 𝒞×𝒟\mathcal{C} \times \mathcal{D} is the [[topologically enriched category]] whose [[objects]] are [[pairs]] of objects (c,d)(c,d) with c𝒞c \in \mathcal{C} and d𝒟d\in \mathcal{D}, whose [[hom-spaces]] are the [[smash product]] of the separate hom-spaces

    (𝒞×𝒟)((c 1,d 1),(c 2,d 2))𝒞(c 1,c 2)𝒟(d 1,d 2) (\mathcal{C}\times \mathcal{D})((c_1,d_1),\;(c_2,d_2) ) \coloneqq \mathcal{C}(c_1,c_2)\wedge \mathcal{D}(d_1,d_2)

    and whose [[composition]] operation is the [[braiding]] followed by the [[smash product]] of the separate composition operations:

    (𝒞×𝒟)((c 1,d 1),(c 2,d 2))(𝒞×𝒟)((c 2,d 2),(c 3,d 3)) = (𝒞(c 1,c 2)𝒟(d 1,d 2))(𝒞(c 2,c 3)𝒟(d 2,d 3)) τ (𝒞(c 1,c 2)𝒞(c 2,c 3))(𝒟(d 1,d 2)𝒟(d 2,d 3)) ( c 1,c 2,c 3)( d 1,d 2,d 3) 𝒞(c 1,c 3)𝒟(d 1,d 3) = (𝒞×𝒟)((c 1,d 1),(c 3,d 3)). \array{ (\mathcal{C}\times \mathcal{D})((c_1,d_1), \; (c_2,d_2)) \wedge (\mathcal{C}\times \mathcal{D})((c_2,d_2), \; (c_3,d_3)) \\ {}^{\mathllap{=}}\downarrow \\ \left(\mathcal{C}(c_1,c_2) \wedge \mathcal{D}(d_1,d_2)\right) \wedge \left(\mathcal{C}(c_2,c_3) \wedge \mathcal{D}(d_2,d_3)\right) \\ \downarrow^{\mathrlap{\tau}}_{\mathrlap{\simeq}} \\ \left(\mathcal{C}(c_1,c_2)\wedge \mathcal{C}(c_2,c_3)\right) \wedge \left( \mathcal{D}(d_1,d_2)\wedge \mathcal{D}(d_2,d_3)\right) &\overset{ (\circ_{c_1,c_2,c_3})\wedge (\circ_{d_1,d_2,d_3}) }{\longrightarrow} & \mathcal{C}(c_1,c_3)\wedge \mathcal{D}(d_1,d_3) \\ && \downarrow^{\mathrlap{=}} \\ && (\mathcal{C}\times \mathcal{D})((c_1,d_1),\; (c_3,d_3)) } \,.
Example

A pointed [[topologically enriched functor]] (def.) into Top cg */Top^{\ast/}_{cg} (exmpl.) out of a pointed topological [[product category]] as in def.

F:𝒞×𝒟Top cg */ F \;\colon\; \mathcal{C} \times \mathcal{D} \longrightarrow Top^{\ast/}_{cg}

(a “pointed topological [[bifunctor]]”) has component maps of the form

F (c 1,d 1),(c 2,d 2):𝒞(c 1,c 2)𝒟(d 1,d 2)Maps(F 0((c 1,d 1)),F 0((c 2,d 2))) *. F_{(c_1,d_1),(c_2,d_2)} \;\colon\; \mathcal{C}(c_1,c_2) \wedge \mathcal{D}(d_1,d_2) \longrightarrow Maps(F_0((c_1,d_1)),F_0((c_2,d_2)))_\ast \,.

By functoriality and under passing to [[adjuncts]] (cor.) this is equivalent to two commuting [[actions]]

ρ c 1,c 2(d):𝒞(c 1,c 2)F 0((c 1,d))F 0((c 2,d)) \rho_{c_1,c_2}(d) \;\colon\; \mathcal{C}(c_1,c_2) \wedge F_0((c_1,d)) \longrightarrow F_0((c_2,d))

and

ρ d 1,d 2(c):𝒟(d 1,d 2)F 0((c,d 1))F 0((c,d 2)). \rho_{d_1,d_2}(c) \;\colon\; \mathcal{D}(d_1,d_2) \wedge F_0((c,d_1)) \longrightarrow F_0((c,d_2)) \,.

In the special case of a functor out of the [[product category]] of some 𝒞\mathcal{C} with its [[opposite category]] (def. )

F:𝒞 op×𝒞Top cg */ F \;\colon\; \mathcal{C}^{op} \times \mathcal{C} \longrightarrow Top^{\ast/}_{cg}

then this takes the form of a “pullback action” in the first variable

ρ c 2,c 1(d):𝒞(c 1,c 2)F 0((c 2,d))F 0((c 1,d)) \rho_{c_2,c_1}(d) \;\colon\; \mathcal{C}(c_1,c_2) \wedge F_0((c_2,d)) \longrightarrow F_0((c_1,d))

and a “pushforward action” in the second variable

ρ d 1,d 2(c):𝒞(d 1,d 2)F 0((c,d 1))F 0((c,d 2)). \rho_{d_1,d_2}(c) \;\colon\; \mathcal{C}(d_1,d_2) \wedge F_0((c,d_1)) \longrightarrow F_0((c,d_2)) \,.
Definition

Let 𝒞\mathcal{C} be a [[small category|small]] pointed [[topologically enriched category]] (def.), i.e. an [[enriched category]] over (Top cg */,,S 0)(Top_{cg}^{\ast/}, \wedge, S^0) from example . Let

F:𝒞 op×𝒞Top cg */ F \;\colon\; \mathcal{C}^{op} \times \mathcal{C} \longrightarrow Top^{\ast/}_{cg}

be a pointed [[topologically enriched functor]] (def.) out of the pointed topological [[product category]] of 𝒞\mathcal{C} with its [[opposite category]], according to def. .

  1. The [[coend]] of FF, denoted c𝒞F(c,c)\overset{c \in \mathcal{C}}{\int} F(c,c), is the [[coequalizer]] in Top cg *Top_{cg}^{\ast} (prop., exmpl., prop., cor.) of the two actions encoded in FF via example :

    c,d𝒞𝒞(c,d)F(d,c)AAAAAAAAc,dρ (d,c)(c)c,dρ (c,d)(d)c𝒞F(c,c)coeqc𝒞F(c,c). \underset{c,d \in \mathcal{C}}{\coprod} \mathcal{C}(c,d) \wedge F(d,c) \underoverset {\underset{\underset{c,d}{\sqcup} \rho_{(d,c)}(c) }{\longrightarrow}} {\overset{\underset{c,d}{\sqcup} \rho_{(c,d)}(d) }{\longrightarrow}} {\phantom{AAAAAAAA}} \underset{c \in \mathcal{C}}{\coprod} F(c,c) \overset{coeq}{\longrightarrow} \overset{c\in \mathcal{C}}{\int} F(c,c) \,.
  2. The [[end]] of FF, denoted c𝒞F(c,c)\underset{c\in \mathcal{C}}{\int} F(c,c), is the [[equalizer]] in Top cg */Top_{cg}^{\ast/} (prop., exmpl., prop., cor.) of the [[adjuncts]] of the two actions encoded in FF via example :

    c𝒞F(c,c)equc𝒞F(c,c)AAAAAAAAc,dρ˜ (c,d)(c)c,dρ˜ d,c(d)c𝒞Maps(𝒞(c,d),F(c,d)) *. \underset{c\in \mathcal{C}}{\int} F(c,c) \overset{\;\;equ\;\;}{\longrightarrow} \underset{c \in \mathcal{C}}{\prod} F(c,c) \underoverset {\underset{\underset{c,d}{\sqcup} \tilde \rho_{(c,d)}(c) }{\longrightarrow}} {\overset{\underset{c,d}{\sqcup} \tilde\rho_{d,c}(d)}{\longrightarrow}} {\phantom{AAAAAAAA}} \underset{c\in \mathcal{C}}{\prod} Maps\left( \mathcal{C}\left(c,d\right), \; F\left(c,d\right) \right)_\ast \,.
Example

Let GG be a [[topological group]]. Write B(G +)\mathbf{B}(G_+) for the pointed [[topologically enriched category]] that has a single object *\ast, whose single [[hom-space]] is G +G_+ (GG with a basepoint freely adjoined (def.))

B(G +)(*,*)G + \mathbf{B}(G_+)(\ast,\ast) \coloneqq G_+

and whose composition operation is the product operation ()()(-)\cdot(-) in GG under adjoining basepoints (exmpl.)

G +G +(G×G) +(()()) +G +. G_+ \wedge G_+ \simeq (G \times G)_+ \overset{((-)\cdot (-))_+}{\longrightarrow} G_+ \,.

Then a [[topologically enriched functor]]

(X,ρ l):B(G +)Top cg */ (X,\rho_l) \;\colon\; \mathbf{B}(G_+) \longrightarrow Top^{\ast/}_{cg}

is a pointed topological space XF(*)X \coloneqq F(\ast) equipped with a continuous function

ρ l:G +XX \rho_l \;\colon\; G_+ \wedge X \longrightarrow X

satisfying the [[action]] property. Hence this is equivalently a continuous and basepoint-preserving left [[action]] (non-linear [[representation]]) of GG on XX.

The [[opposite category]] (def. ) (B(G +)) op(\mathbf{B}(G_+))^{op} comes from the [[opposite group]]

(B(G +)) op=B(G + op). (\mathbf{B}(G_+))^{op} = \mathbf{B}(G^{op}_+) \,.

(The canonical continuous isomorphism GG opG \simeq G^{op} induces a canonical euqivalence of topologically enriched categories (B(G +)) opB(G +)(\mathbf{B}(G_+))^{op} \simeq \mathbf{B}(G_+).)

So a topologically enriched functor

(Y,ρ r):(B(G +)) opTop cg * (Y,\rho_r) \;\colon\; (\mathbf{B}(G_+))^{op} \longrightarrow Top^{\ast}_{cg}

is equivalently a basepoint preserving continuous right action of GG.

Therefore the [[coend]] of two such functors (def. ) coequalizes the relation

(xg,y)(x,gy) (x g,\;y) \sim (x,\; g y)

(where juxtaposition denotes left/right action) and hence is equivalently the canonical smash product of a right GG-action with a left GG-action, hence the [[quotient topological space|quotient]] of the plain smash product by the [[diagonal action]] of the group GG:

*B(G +)(Y,ρ r)(*)(X,ρ l)(*)Y GX. \overset{\ast \in \mathbf{B}(G_+)}{\int} (Y,\rho_r)(\ast) \,\wedge\, (X,\rho_l)(\ast) \;\simeq\; Y \wedge_G X \,.
Example

Let 𝒞\mathcal{C} be a [[small category|small]] pointed [[topologically enriched category]] (def.). For F,G:𝒞Top cg */ F,G \;\colon\; \mathcal{C} \longrightarrow Top^{\ast/}_{cg} two pointed [[topologically enriched functors]], then the [[end]] (def. ) of Maps(F(),G()) *Maps(F(-),G(-))_\ast is a topological space whose underlying [[pointed set]] is the pointed set of [[natural transformations]] FGF\to G (def.):

U(c𝒞Maps(F(c),G(c)) *)Hom [𝒞,Top cg */](F,G). U \left( \underset{c \in \mathcal{C}}{\int} Maps(F(c),G(c))_\ast \right) \;\simeq\; Hom_{[\mathcal{C},Top^{\ast/}_{cg}]}(F,G) \,.
Proof

The underlying pointed set functor U:Top cg */Set */U\colon Top^{\ast/}_{cg}\to Set^{\ast/} [[preserved limit|preserves]] all [[limits]] (prop., prop., prop.). Therefore there is an [[equalizer]] diagram in Set */Set^{\ast/} of the form

U(c𝒞Maps(F(c),G(c)) *)equc𝒞Hom Top cg */(F(c),G(c))AAAAAAAAc,dU(ρ˜ (c,d)(c))c,dU(ρ˜ d,c(d))c,d𝒞Hom Top cg */(𝒞(c,d),Maps(F(c),G(d)) *). U \left( \underset{c\in \mathcal{C}}{\int} Maps(F(c),G(c))_\ast \right) \overset{equ}{\longrightarrow} \underset{c\in \mathcal{C}}{\prod} Hom_{Top^{\ast/}_{cg}}(F(c),G(c)) \underoverset {\underset{\underset{c,d}{\sqcup} U(\tilde \rho_{(c,d)}(c)) }{\longrightarrow}} {\overset{\underset{c,d}{\sqcup} U(\tilde\rho_{d,c}(d))}{\longrightarrow}} {\phantom{AAAAAAAA}} \underset{c,d\in \mathcal{C}}{\prod} Hom_{Top^{\ast/}_{cg}}( \mathcal{C}(c,d), Maps(F(c),G(d))_\ast ) \,.

Here the object in the middle is just the set of collections of component morphisms {F(c)η cG(c)} c𝒞\left\{ F(c)\overset{\eta_c}{\to} G(c)\right\}_{c\in \mathcal{C}}. The two parallel maps in the equalizer diagram take such a collection to the functions which send any cfdc \overset{f}{\to} d to the result of precomposing

F(c) f(f) F(d) η d G(d) \array{ F(c) \\ {}^{\mathllap{f(f)}}\downarrow \\ F(d) &\underset{\eta_d}{\longrightarrow}& G(d) }

and of postcomposing

F(c) η c G(c) G(f) G(d) \array{ F(c) &\overset{\eta_c}{\longrightarrow}& G(c) \\ && \downarrow^{\mathrlap{G(f)}} \\ && G(d) }

each component in such a collection, respectively. These two functions being equal, hence the collection {η c} c𝒞\{\eta_c\}_{c\in \mathcal{C}} being in the equalizer, means precisley that for all c,dc,d and all f:cdf\colon c \to d the square

F(c) η c G(c) F(f) G(f) F(d) η d G(g) \array{ F(c) &\overset{\eta_c}{\longrightarrow}& G(c) \\ {}^{\mathllap{F(f)}}\downarrow && \downarrow^{\mathrlap{G(f)}} \\ F(d) &\underset{\eta_d}{\longrightarrow}& G(g) }

is a [[commuting square]]. This is precisley the condition that the collection {η c} c𝒞\{\eta_c\}_{c\in \mathcal{C}} be a [[natural transformation]].

Conversely, example says that [[ends]] over [[bifunctors]] of the form Maps(F(),G())) *Maps(F(-),G(-)))_\ast constitute [[hom-spaces]] between pointed [[topologically enriched functors]]:

Definition

Let 𝒞\mathcal{C} be a [[small category|small]] pointed [[topologically enriched category]] (def.). Define the structure of a pointed [[topologically enriched category]] on the category [𝒞,Top cg */][\mathcal{C}, Top_{cg}^{\ast/}] of pointed [[topologically enriched functors]] to Top cg */Top^{\ast/}_{cg} (exmpl.) by taking the [[hom-spaces]] to be given by the [[ends]] (def. ) of example :

[𝒞,Top cg */](F,G) c𝒞Maps(F(c),G(c)) * [\mathcal{C},Top^{\ast/}_{cg}](F,G) \;\coloneqq\; \int_{c\in \mathcal{C}} Maps(F(c),G(c))_\ast

The [[composition]] operation on these is defined to be the one induced by the composite maps

(c𝒞Maps(F(c),G(c)) *)(c𝒞Maps(G(c),H(c)) *)c𝒞Maps(F(c),G(c)) *Maps(G(c),H(c)) *( F(c),G(c),H(c)) c𝒞c𝒞Maps(F(c),H(c)) *, \left( \underset{c\in \mathcal{C}}{\int} Maps(F(c),G(c))_\ast \right) \wedge \left( \underset{c \in \mathcal{C}}{\int} Maps(G(c),H(c))_\ast \right) \overset{}{\longrightarrow} \underset{c\in \mathcal{C}}{\prod} Maps(F(c),G(c))_\ast \wedge Maps(G(c),H(c))_\ast \overset{(\circ_{F(c),G(c),H(c)})_{c\in \mathcal{C}}}{\longrightarrow} \underset{c \in \mathcal{C}}{\prod} Maps(F(c),H(c))_\ast \,,

where the first, morphism is degreewise given by projection out of the limits that defined the ends. This composite evidently equalizes the two relevant adjunct actions (as in the proof of example ) and hence defines a map into the end

(c𝒞Maps(F(c),G(c)) *)(c𝒞Maps(G(c),H(c)) *)c𝒞Maps(F(c),H(c)) *. \left( \underset{c\in \mathcal{C}}{\int} Maps(F(c),G(c))_\ast \right) \wedge \left( \underset{c \in \mathcal{C}}{\int} Maps(G(c),H(c))_\ast \right) \longrightarrow \underset{c\in \mathcal{C}}{\int} Maps(F(c),H(c))_\ast \,.

The resulting pointed [[topologically enriched category]] [𝒞,Top cg */][\mathcal{C},Top^{\ast/}_{cg}] is also called the Top cg */Top^{\ast/}_{cg}-[[enriched functor category]] over 𝒞\mathcal{C} with coefficients in Top cg */Top^{\ast/}_{cg}.

This yields an equivalent formulation in terms of ends of the pointed topologically [[enriched Yoneda lemma]] (prop.):

Proposition

(topologically [[enriched Yoneda lemma]])

Let 𝒞\mathcal{C} be a [[small category|small]] pointed [[topologically enriched categories]] (def.). For F:𝒞Top cg */F \colon \mathcal{C}\to Top^{\ast/}_{cg} a pointed [[topologically enriched functor]] (def.) and for c𝒞c\in \mathcal{C} an object, there is a [[natural isomorphism]]

[𝒞,Top cg */](𝒞(c,),F)F(c) [\mathcal{C}, Top^{\ast/}_{cg}](\mathcal{C}(c,-),\; F) \;\simeq\; F(c)

between the [[hom-space]] of the pointed topological functor category, according to def. , from the [[representable functor|functor represented]] by cc to FF, and the value of FF on cc.

In terms of the [[ends]] (def. ) defining these [[hom-spaces]], this means that

d𝒞Maps(𝒞(c,d),F(d)) *F(c). \underset{d\in \mathcal{C}}{\int} Maps(\mathcal{C}(c,d), F(d))_\ast \;\simeq\; F(c) \,.

In this form the statement is also known as [[Yoneda reduction]].

The proof of prop. is [[formal dual|formally dual]] to the proof of the next prop. .

Now that [[natural transformations]] are expressed in terms of [[ends]] (example ), as is the Yoneda lemma (prop. ), it is natural to consider the [[formal duality|dual]] statement involving [[coends]]:

Proposition

([[co-Yoneda lemma]])

Let 𝒞\mathcal{C} be a [[small category|small]] pointed [[topologically enriched category]] (def.). For F:𝒞Top cg */F \colon \mathcal{C}\to Top^{\ast/}_{cg} a pointed [[topologically enriched functor]] (def.) and for c𝒞c\in \mathcal{C} an object, there is a [[natural isomorphism]]

F()c𝒞𝒞(c,)F(c). F(-) \simeq \overset{c \in \mathcal{C}}{\int} \mathcal{C}(c,-) \wedge F(c) \,.

Moreover, the morphism that hence exhibits F(c)F(c) as the [[coequalizer]] of the two morphisms in def. is componentwise the canonical action

𝒞(c,d)F(c)F(d) \mathcal{C}(c,d) \wedge F(c) \longrightarrow F(d)

which is [[adjunct]] to the component map 𝒞(d,c)Maps(F(c),F(d)) *\mathcal{C}(d,c) \to Maps(F(c),F(d))_{\ast} of the [[topologically enriched functor]] FF.

(e.g. MMSS 00, lemma 1.6)

Proof

The coequalizer of pointed topological spaces that we need to consider has underlying it a coequalizer of underlying pointed sets (prop., prop., prop.). That in turn is the colimit over the diagram of underlying sets with the basepointe adjoined to the diagram (prop.). For a coequalizer diagram adding that extra point to the diagram clearly does not change the colimit, and so we need to consider the plain coequalizer of sets.

That is just the set of [[equivalence classes]] of [[pairs]]

(cc 0,x)𝒞(c,c 0)F(c), ( c \overset{}{\to} c_0,\; x ) \;\; \in \mathcal{C}(c,c_0) \wedge F(c) \,,

where two such pairs

(cfc 0,xF(c)),(dgc 0,yF(d)) ( c \overset{f}{\to} c_0,\; x \in F(c) ) \,,\;\;\;\; ( d \overset{g}{\to} c_0,\; y \in F(d) )

are regarded as equivalent if there exists

cϕd c \overset{\phi}{\to} d

such that

f=gϕ,andy=ϕ(x). f = g \circ \phi \,, \;\;\;\;\;and\;\;\;\;\; y = \phi(x) \,.

(Because then the two pairs are the two images of the pair (g,x)(g,x) under the two morphisms being coequalized.)

But now considering the case that d=c 0d = c_0 and g=id c 0g = id_{c_0}, so that f=ϕf = \phi shows that any pair

(cϕc 0,xF(c)) ( c \overset{\phi}{\to} c_0, \; x \in F(c))

is identified, in the coequalizer, with the pair

(id c 0,ϕ(x)F(c 0)), (id_{c_0},\; \phi(x) \in F(c_0)) \,,

hence with ϕ(x)F(c 0)\phi(x)\in F(c_0).

This shows the claim at the level of the underlying sets. To conclude it is now sufficient (prop.) to show that the topology on F(c 0)Top cg */F(c_0) \in Top^{\ast/}_{cg} is the [[final topology]] (def.) of the system of component morphisms

𝒞(d,c)F(c)c𝒞(c,c 0)F(c) \mathcal{C}(d,c) \wedge F(c) \longrightarrow \overset{c}{\int} \mathcal{C}(c,c_0) \wedge F(c)

which we just found. But that system includes

𝒞(c,c)F(c)F(c) \mathcal{C}(c,c) \wedge F(c) \longrightarrow F(c)

which is a [[retraction]]

id:F(c)𝒞(c,c)F(c)F(c) id \;\colon\; F(c) \longrightarrow \mathcal{C}(c,c) \wedge F(c) \longrightarrow F(c)

and so if all the preimages of a given subset of the coequalizer under these component maps is open, it must have already been open in F(c)F(c).

Remark

The statement of the [[co-Yoneda lemma]] in prop. is a kind of [[categorification]] of the following statement in [[analysis]] (whence the notation with the integral signs):

For XX a [[topological space]], f:Xf \colon X \to\mathbb{R} a [[continuous function]] and δ(,x 0)\delta(-,x_0) denoting the [[Dirac distribution]], then

xXδ(x,x 0)f(x)=f(x 0). \int_{x \in X} \delta(x,x_0) f(x) = f(x_0) \,.

It is this analogy that gives the name to the following statement:

Proposition

([[Fubini theorem]] for (co)-ends)

For FF a pointed topologically enriched [[bifunctor]] on a small pointed topological [[product category]] 𝒞 1×𝒞 2\mathcal{C}_1 \times \mathcal{C}_2 (def. ), i.e.

F:(𝒞 1×𝒞 2) op×(𝒞 1×𝒞 2)Top cg */ F \;\colon\; \left( \mathcal{C}_1\times\mathcal{C}_2 \right)^{op} \times (\mathcal{C}_1 \times\mathcal{C}_2) \longrightarrow Top^{\ast/}_{cg}

then its [[end]] and [[coend]] (def. ) is equivalently formed consecutively over each variable, in either order:

(c 1,c 2)F((c 1,c 2),(c 1,c 2))c 1c 2F((c 1,c 2),(c 1,c 2))c 2c 1F((c 1,c 2),(c 1,c 2)) \overset{(c_1,c_2)}{\int} F((c_1,c_2), (c_1,c_2)) \simeq \overset{c_1}{\int} \overset{c_2}{\int} F((c_1,c_2), (c_1,c_2)) \simeq \overset{c_2}{\int} \overset{c_1}{\int} F((c_1,c_2), (c_1,c_2))

and

(c 1,c 2)F((c 1,c 2),(c 1,c 2))c 1c 2F((c 1,c 2),(c 1,c 2))c 2c 1F((c 1,c 2),(c 1,c 2)). \underset{(c_1,c_2)}{\int} F((c_1,c_2), (c_1,c_2)) \simeq \underset{c_1}{\int} \underset{c_2}{\int} F((c_1,c_2), (c_1,c_2)) \simeq \underset{c_2}{\int} \underset{c_1}{\int} F((c_1,c_2), (c_1,c_2)) \,.
Proof

Because [[limits]] commute with limits, and [[colimits]] commute with colimits.

Remark

Since the pointed compactly generated [[mapping space]] functor (exmpl.)

Maps(,) *:(Top cg */) op×Top cg */Top cg */ Maps(-,-)_\ast \;\colon\; \left(Top^{\ast/}_{cg}\right)^{op} \times Top^{\ast/}_{cg} \longrightarrow Top^{\ast/}_{cg}

takes [[colimits]] in the first argument and [[limits]] in the second argument to limits (cor.), it in particular takes [[coends]] in the first argument and [[ends]] in the second argument, to ends (def. ):

Maps(X, cF(c,c)) * cMaps(X,F(c,c) *) Maps( X, \; \int_{c} F(c,c))_\ast \simeq \int_c Maps(X, F(c,c)_\ast)

and

Maps( cF(c,c),Y) *cMaps(F(c,c),Y) *. Maps( \int^{c} F(c,c),\; Y )_\ast \simeq \underset{c}{\int} Maps( F(c,c),\; Y )_\ast \,.

With this [[coend]] calculus in hand, there is an elegant proof of the defining [[universal property]] of the smash [[tensoring]] of [[topologically enriched functors]] [𝒞,Top cg *][\mathcal{C},Top^{\ast}_{cg}] (def.)

Proposition

For 𝒞\mathcal{C} a pointed [[topologically enriched category]], there are [[natural isomorphisms]]

[𝒞,Top cg */](XK,Y)Maps(K,[𝒞,Top cg */](X,Y)) * [\mathcal{C},Top^{\ast/}_{cg}]( X \wedge K ,\, Y ) \;\simeq\; Maps(K,\; [\mathcal{C},Top^{\ast/}_{cg}](X,Y))_\ast

and

[𝒞,Top cg */](X,Maps(K,Y) *)Maps(K,[𝒞,Top cg */](X,Y)) [\mathcal{C},Top^{\ast/}_{cg}](X,\, Maps(K,Y)_\ast) \;\simeq\; Maps(K,\; [\mathcal{C},Top^{\ast/}_{cg}](X,Y))

for all X,Y[𝒞,Top cg */]X,Y \in [\mathcal{C},Top^{\ast/}_{cg}] and all KTop cg */K \in Top^{\ast/}_{cg}.

In particular there is the combined natural isomorphism

[𝒞,Top cg */](XK,Y)[𝒞,Top cg */](X,Maps(K,Y) *) [\mathcal{C}, Top^{\ast/}_{cg}](X\wedge K, Y) \;\simeq\; [\mathcal{C}, Top^{\ast/}_{cg}](X, Maps(K,Y)_\ast)

exhibiting a pair of [[adjoint functors]]

[𝒞,Top cg */]Maps(K,) *()K[𝒞,Top cg *]. [\mathcal{C}, Top^{\ast/}_{cg}] \underoverset {\underset{Maps(K,-)_\ast}{\longrightarrow}} {\overset{(-)\wedge K}{\longleftarrow}} {\bot} [\mathcal{C}, Top^{\ast}_{cg}] \,.
Proof

Via the [[end]]-expression for [𝒞,Top cg */](,)[\mathcal{C},Top^{\ast/}_{cg}](-,-) from def. and the fact (remark ) that the pointed mapping space construction Maps(,) *Maps(-,-)_\ast preserves ends in the second variable, this reduces to the fact that Maps(,) *Maps(-,-)_\ast is the [[internal hom]] in the [[closed monoidal category]] Top cg */Top^{\ast/}_{cg} (example ) and hence satisfies the internal tensor/hom-adjunction isomorphism (prop. ):

[𝒞,Top cg */](XK,Y) =cMaps((XK)(c),Y(c)) * cMaps(X(c)K,Y(x)) * cMaps(K,Maps(X(c),Y(c)) *) * Maps(K,cMaps(X(c),Y(c))) * =Maps(K,[𝒞,Top cg */](X,Y)) * \begin{aligned} [\mathcal{C},Top^{\ast/}_{cg}](X \wedge K, Y) & = \underset{c}{\int} Maps( (X \wedge K)(c), Y(c) )_\ast \\ & \simeq \underset{c}{\int} Maps(X(c) \wedge K, Y(x))_\ast \\ & \simeq \underset{c}{\int} Maps(K,Maps(X(c), Y(c))_\ast)_\ast \\ & \simeq Maps(K, \underset{c}{\int} Maps(X(c),Y(c)))_\ast \\ & = Maps(K,[\mathcal{C},Top^{\ast/}_{cg}](X,Y))_\ast \end{aligned}

and

[𝒞,Top cg */](X,Maps(K,Y) *) =cMaps(X(c),(Maps(K,Y) *)(c)) * cMaps(X(c),Maps(K,Y(c)) *) * cMaps(X(c)K,Y(c)) * cMaps(K,Maps(X(c),Y(c)) *) * Maps(K,cMaps(X(c),Y(c)) *) * Maps(K,[𝒞,Top cg */](X,Y)) *. \begin{aligned} [\mathcal{C},Top^{\ast/}_{cg}](X, Maps(K,Y)_\ast) & = \underset{c}{\int} Maps(X(c), (Maps(K,Y)_\ast)(c))_\ast \\ & \simeq \underset{c}{\int} Maps(X(c), Maps(K,Y(c))_\ast)_\ast \\ & \simeq \underset{c}{\int} Maps(X(c) \wedge K, Y(c))_\ast \\ & \simeq \underset{c}{\int} Maps(K, Maps(X(c),Y(c))_\ast)_\ast \\ & \simeq Maps(K, \underset{c}{\int} Maps(X(c),Y(c))_\ast)_\ast \\ & \simeq Maps(K, [\mathcal{C},Top^{\ast/}_{cg}](X,Y))_\ast \,. \end{aligned}
Proposition

(left Kan extension via coends)

Let 𝒞,𝒟\mathcal{C}, \mathcal{D} be [[small category|small]] pointed [[topologically enriched categories]] (def.) and let

p:𝒞𝒟 p \;\colon\; \mathcal{C} \longrightarrow \mathcal{D}

be a pointed [[topologically enriched functor]] (def.). Then precomposition with pp constitutes a functor

p *:[𝒟,Top cg */][𝒞,Top cg */] p^\ast \;\colon\; [\mathcal{D}, Top^{\ast/}_{cg}] \longrightarrow [\mathcal{C}, Top^{\ast/}_{cg}]

GGpG\mapsto G\circ p. This functor has a [[left adjoint]] Lan pLan_p, called left [[Kan extension]] along pp

[𝒟,Top cg */]p *Lan p[𝒞,Top cg */] [\mathcal{D}, Top^{\ast/}_{cg}] \underoverset {\underset{p^\ast}{\longrightarrow}} {\overset{Lan_p }{\longleftarrow}} {\bot} [\mathcal{C}, Top^{\ast/}_{cg}]

which is given objectwise by a [[coend]] (def. ):

(Lan pF):dc𝒞𝒟(p(c),d)F(c). (Lan_p F) \;\colon\; d \;\mapsto \; \overset{c\in \mathcal{C}}{\int} \mathcal{D}(p(c),d) \wedge F(c) \,.
Proof

Use the expression of natural transformations in terms of ends (example and def. ), then use the respect of Maps(,) *Maps(-,-)_\ast for ends/coends (remark ), use the smash/mapping space adjunction (cor.), use the [[Fubini theorem]] (prop. ) and finally use [[Yoneda reduction]] (prop. ) to obtain a sequence of [[natural isomorphisms]] as follows:

[𝒟,Top cg */](Lan pF,G) =d𝒟Maps((Lan pF)(d),G(d)) * =d𝒟Maps(c𝒞𝒟(p(c),d)F(c),G(d)) * d𝒟c𝒞Maps(𝒟(p(c),d)F(c),G(d)) * c𝒞d𝒟Maps(F(c),Maps(𝒟(p(c),d),G(d)) *) * c𝒞Maps(F(c),d𝒟Maps(𝒟(p(c),d),G(d)) *) * c𝒞Maps(F(c),G(p(c))) * =[𝒞,Top cg */](F,p *G). \begin{aligned} [\mathcal{D},Top^{\ast/}_{cg}]( Lan_p F, \, G ) & = \underset{d \in \mathcal{D}}{\int} Maps( (Lan_p F)(d), \, G(d) )_\ast \\ & = \underset{d\in \mathcal{D}}{\int} Maps\left( \overset{c \in \mathcal{C}}{\int} \mathcal{D}(p(c),d) \wedge F(c) ,\; G(d) \right)_\ast \\ &\simeq \underset{d \in \mathcal{D}}{\int} \underset{c \in \mathcal{C}}{\int} Maps( \mathcal{D}(p(c),d)\wedge F(c) \,,\; G(d) )_\ast \\ & \simeq \underset{c\in \mathcal{C}}{\int} \underset{d\in \mathcal{D}}{\int} Maps(F(c), Maps( \mathcal{D}(p(c),d) , \, G(d) )_\ast )_\ast \\ & \simeq \underset{c\in \mathcal{C}}{\int} Maps(F(c), \underset{d\in \mathcal{D}}{\int} Maps( \mathcal{D}(p(c),d) , \, G(d) )_\ast )_\ast \\ & \simeq \underset{c\in \mathcal{C}}{\int} Maps(F(c), G(p(c)) )_\ast \\ & = [\mathcal{C}, Top^{\ast/}_{cg}](F,p^\ast G) \end{aligned} \,.
Topological Day convolution

Given two [[functions]] f 1,f 2:Gf_1, f_2 \colon G \longrightarrow \mathbb{C} on a [[group]] (or just a [[monoid]]) GG, then their [[convolution product]] is, whenever well defined, given by the sum

f 1f 2:gg 1g 2=gf 1(g 1)f 2(g 2). f_1 \star f_2 \;\colon\; g \mapsto \underset{g_1 \cdot g_2 = g}{\sum} f_1(g_1) \cdot f_2(g_2) \,.

The operation of [[Day convolution]] is the [[categorification]] of this situation where functions are replaced by [[functors]] and [[monoids]] by [[monoidal categories]]. Further below we find the [[symmetric monoidal smash product of spectra]] as the Day convolution of topologically enriched functors over the monoidal category of finite pointed CW-complexes, or over sufficiently rich subcategories thereof.

Definition

Let (𝒞,,1)(\mathcal{C}, \otimes, 1) be a [[small category|small]] pointed [[topologically enriched category|topological]] [[monoidal category]] (def. ).

Then the [[Day convolution]] tensor product on the pointed topological [[enriched functor category]] [𝒞,Top cg */][\mathcal{C},Top^{\ast/}_{cg}] (def. ) is the [[functor]]

Day:[𝒞,Top cg */]×[𝒞,Top cg */][𝒞,Top cg */] \otimes_{Day} \;\colon\; [\mathcal{C},Top^{\ast/}_{cg}] \times [\mathcal{C},Top^{\ast/}_{cg}] \longrightarrow [\mathcal{C},Top^{\ast/}_{cg}]

out of the pointed topological [[product category]] (def. ) given by the following [[coend]] (def. )

X DayY:c(c 1,c 2)𝒞×𝒞𝒞(c 1c 2,c)X(c 1)Y(c 2). X \otimes_{Day} Y \;\colon\; c \;\mapsto\; \overset{(c_1,c_2)\in \mathcal{C}\times \mathcal{C}}{\int} \mathcal{C}(c_1 \otimes c_2, c) \wedge X(c_1) \wedge Y(c_2) \,.
Example

Let SeqSeq denote the category with objects the [[natural numbers]], and only the [[zero morphisms]] and [[identity morphisms]] on these objects (we consider this in a braoder context below in def. ):

Seq(n 1,n 2){S 0 ifn 1=n 2 * otherwise. Seq(n_1,n_2) \coloneqq \left\{ \array{ S^0 & if\; n_1 = n_2 \\ \ast & otherwise } \right. \,.

Regard this as a pointed topologically enriched category in the unique way. The operation of addition of natural numbers =+\otimes = + makes this a monoidal category.

An object X [Seq,Top cg */]X_\bullet \in [Seq, Top_{cg}^{\ast/}] is an \mathbb{N}-sequence of pointed topological spaces. Given two such, then their Day convolution according to def. is

(X DayY) n =(n 1,n 2)Seq(n 1+n 2,n)X n 1X n 2 =n 1+n 2=n(X n 1X n 2). \begin{aligned} (X \otimes_{Day} Y)_n & = \overset{(n_1,n_2)}{\int} Seq(n_1 + n_2 , n) \wedge X_{n_1} \wedge X_{n_2} \\ & = \underset{{n_1+n_2} \atop {= n}}{\coprod} \left(X_{n_1}\wedge X_{n_2}\right) \end{aligned} \,.

We observe now that [[Day convolution]] is equivalently a [[left Kan extension]] (def. ). This will be key for understanding [[monoids]] and [[modules]] with respect to Day convolution.

Definition

Let 𝒞\mathcal{C} be a [[small category|small]] pointed [[topologically enriched category]] (def.). Its [[external tensor product]] is the pointed [[topologically enriched functor]]

¯:[𝒞,Top cg */]×[𝒞,Top cg */][𝒞×𝒞,Top cg */] \overline{\wedge} \;\colon\; [\mathcal{C},Top^{\ast/}_{cg}] \times [\mathcal{C},Top^{\ast/}_{cg}] \longrightarrow [\mathcal{C}\times \mathcal{C}, Top^{\ast/}_{cg}]

from pairs of [[topologically enriched functors]] over mmathcalC\mmathcal{C} to topologically enriched functors over the [[product category]] 𝒞×𝒞\mathcal{C} \times \mathcal{C} (def. ) given by

X¯Y(X,Y), X \overline{\wedge} Y \;\coloneqq\; \wedge \circ (X,Y) \,,

i.e.

(X¯Y)(c 1,c 2)=X(c 1)X(c 2). (X \overline\wedge Y)(c_1,c_2) = X(c_1)\wedge X(c_2) \,.
Proposition

For (𝒞,1)(\mathcal{C}, \otimes 1) a pointed [[topologically enriched category|topologically enriched]] [[monoidal category]] (def. ) the [[Day convolution]] product (def. ) of two functors is equivalently the [[left Kan extension]] (def. ) of their external tensor product (def. ) along the tensor product :𝒞×𝒞\otimes \colon \mathcal{C} \times \mathcal{C}: there is a [[natural isomorphism]]

X DayYLan (X¯Y). X \otimes_{Day} Y \simeq Lan_{\otimes} (X \overline{\wedge} Y) \,.

Hence the [[adjunction unit]] is a [[natural transformation]] of the form

𝒞×𝒞 X¯Y Top cg */ X DayY 𝒞. \array{ \mathcal{C} \times \mathcal{C} && \overset{X \overline{\wedge} Y}{\longrightarrow} && Top^{\ast/}_{cg} \\ & {}^{\mathllap{\otimes}}\searrow &\Downarrow& \nearrow_{\mathrlap{X \otimes_{Day} Y}} \\ && \mathcal{C} } \,.

This perspective is highlighted in (MMSS 00, p. 60).

Proof

By prop. we may compute the left Kan extension as the following [[coend]]:

Lan 𝒞(X¯Y)(c) (c 1,c 2)𝒞(c 1 𝒞c 2,c)(X¯Y)(c 1,c 2) =(c 1,c 2)𝒞(c 1c 2,c)X(c 1)X(c 2). \begin{aligned} Lan_{\otimes_{\mathcal{C}}} (X\overline{\wedge} Y)(c) & \simeq \overset{(c_1,c_2)}{\int} \mathcal{C}(c_1 \otimes_{\mathcal{C}} c_2, c ) \wedge (X\overline{\wedge}Y)(c_1,c_2) \\ & = \overset{(c_1,c_2)}{\int} \mathcal{C}(c_1\otimes c_2, c) \wedge X(c_1)\wedge X(c_2) \end{aligned} \,.

Proposition implies the following fact, which is the key for the identification of “[[functors with smash product]]” below and then for the description of [[ring spectra]] further below.

Corollary

The operation of [[Day convolution]] Day\otimes_{Day} (def. ) is universally characterized by the property that there are [[natural isomorphisms]]

[𝒞,Top cg */](X DayY,Z)[𝒞×𝒞,Top cg */](X¯Y,Z), [\mathcal{C},Top^{\ast/}_{cg}](X \otimes_{Day} Y, Z) \simeq [\mathcal{C}\times \mathcal{C},Top^{\ast/}_{cg}]( X \overline{\wedge} Y,\; Z \circ \otimes ) \,,

where ¯\overline{\wedge} is the external product of def. , hence that [[natural transformations]] of functors on 𝒞\mathcal{C} of the form

(X DayY)(c)Z(c) (X \otimes_{Day} Y)(c) \longrightarrow Z(c)

are in [[natural bijection]] with natural transformations of functors on the [[product category]] mmathcalC×𝒞\mmathcal{C}\times \mathcal{C} (def. ) of the form

X(c 1)Y(c 2)Z(c 1c 2). X(c_1) \wedge Y(c_2) \longrightarrow Z(c_1 \otimes c_2) \,.

Write

y:𝒞 op[𝒞,Top cg */] y \;\colon\; \mathcal{C}^{op} \longrightarrow [\mathcal{C}, Top^{\ast/}_{cg}]

for the Top cg */Top^{\ast/}_{cg}-[[Yoneda embedding]], so that for c𝒞c\in \mathcal{C} any [[object]], y(c)y(c) is the [[representable functor|corepresented functor]] y(c):d𝒞(c,d)y(c)\colon d \mapsto \mathcal{C}(c,d).

Proposition

For (𝒞,,1)(\mathcal{C},\otimes, 1) a [[small category|small]] pointed [[topologically enriched category|topological]] [[monoidal category]] (def. ), the [[Day convolution]] tensor product Day\otimes_{Day} of def. makes the pointed topologically [[enriched functor category]]

([𝒞,Top cg */], Day,y(1)) ( [\mathcal{C}, Top^{\ast/}_{cg}], \otimes_{Day}, y(1))

into a pointed topological [[monoidal category]] (def. ) with [[tensor unit]] y(1)y(1) [[representable functor|co-represented]] by the tensor unit 11 of 𝒞\mathcal{C}.

Moreover, if (𝒞,,1)(\mathcal{C}, \otimes, 1) is equipped with a (symmetric) [[braiding]] τ 𝒞\tau^{\mathcal{C}} (def. ), then so is ([𝒞,Top cg */], Day,y(1))([\mathcal{C}, Top^{\ast/}_{cg}],\otimes_{Day}, y(1)).

Proof

Regarding [[associativity]], observe that

(X Day(Y DayZ))(c) (c 1,c 2)𝒞(c 1c 2,c)X(c 1)(d 1,d 2)𝒞(d 1d 2,c 2)(Y(d 1)Z(d 2)) c 1,d 1,d 2c 2𝒞(c 1c 2,c)𝒞(d 1d 2,c 2)𝒞(c 1(d 1 𝒞d 2),c)(X(c 1)(Y(d 1)Z(d 2))) c 1,d 1,d 2𝒞(c 1(d 1d 2),c)(X(c 1)(Y(d 1)Z(d 2))) c 1,c 2,c 3𝒞(c 1(c 2c 3),c)(X(c 1)(Y(c 2)Z(c 3))), \begin{aligned} (X \otimes_{Day} ( Y \otimes_{Day} Z ))(c) & \simeq \overset{(c_1,c_2)}{\int} \mathcal{C}(c_1 \otimes c_2, \,c) \wedge X(c_1) \wedge \overset{(d_1,d_2)}{\int} \mathcal{C}(d_1 \otimes d_2, c_2 ) (Y(d_1) \wedge Z(d_2)) \\ &\simeq \overset{c_1, d_1, d_2}{\int} \underset{\simeq \mathcal{C}(c_1 \otimes (d_1 \otimes_{\mathcal{C}} d_2), c )}{ \underbrace{ \overset{c_2}{\int} \mathcal{C}(c_1 \otimes c_2 , c) \wedge \mathcal{C}(d_1 \otimes d_2, c_2 ) } } \wedge (X(c_1) \wedge (Y(d_1) \wedge Z(d_2))) \\ &\simeq \overset{c_1, d_1, d_2}{\int} \mathcal{C}(c_1\otimes ( d_1 \otimes d_2), c ) \wedge (X(c_1) \wedge (Y(d_1) \wedge Z(d_2))) \\ & \simeq \overset{c_1, c_2, c_3}{\int} \mathcal{C}(c_1\otimes ( c_2 \otimes c_3), c ) \wedge (X(c_1) \wedge (Y(c_2) \wedge Z(c_3))) \end{aligned} \,,

where we used the [[Fubini theorem]] for [[coends]] (prop. ) and then twice the [[co-Yoneda lemma]] (prop. ). Similarly

((X DayY) DayZ)(c) (c 1,c 2)𝒞(c 1c 2,c)(d 1,d 2)𝒞(d 1d 2,c 1)(X(d 1)Y(d 2))Y(c 2) c 2,d 1,d 2c 1𝒞(c 1c 2,c)𝒞(d 1d 2,c 1)𝒞((d 1d 2)c 2)((X(d 1)Y(d 2))Z(c 2)) c 2,d 1,d 2𝒞((d 1d 2)c 2)((X(d 1)Y(d 2))Z(c 2)) c 1,c 2,c 3𝒞((c 1c 2)c 3)((X(c 1)Y(c 2))Z(c 3)). \begin{aligned} (( X \otimes_{Day} Y ) \otimes_{Day} Z)(c) & \simeq \overset{(c_1,c_2)}{\int} \mathcal{C}(c_1 \otimes c_2, c) \wedge \overset{(d_1,d_2)}{\int} \mathcal{C}(d_1 \otimes d_2, c_1) \wedge (X(d_1) \wedge Y(d_2)) \wedge Y(c_2) \\ & \simeq \overset{c_2,d_1,d_2}{\int} \underset{\simeq \mathcal{C}((d_1 \otimes d_2) \otimes c_2) }{ \underbrace{ \overset{c_1}{\int} \mathcal{C}(c_1\otimes c_2, c) \wedge \mathcal{C}(d_1 \otimes d_2, c_1) }} \wedge ((X(d_1) \wedge Y(d_2)) \wedge Z(c_2)) \\ & \simeq \overset{c_2,d_1,d_2}{\int} \mathcal{C}((d_1 \otimes d_2) \otimes c_2) \wedge ((X(d_1) \wedge Y(d_2)) \wedge Z(c_2)) \\ &\simeq \overset{c_1,c_2, c_3}{\int} \mathcal{C}((c_1 \otimes c_2) \otimes c_3) \wedge ((X(c_1) \wedge Y(c_2)) \wedge Z(c_3)) \end{aligned} \,.

So we obtain an [[associator]] by combining, in the integrand, the associator α 𝒞\alpha^{\mathcal{C}} of (𝒞,,1)(\mathcal{C}, \otimes, 1) and τ Top cg */\tau^{Top_{cg}^{\ast/}} of (Top cg */,,S 0)(Top^{\ast/}_{cg}, \wedge, S^0) (example ):

((X DayY) DayZ)(c) c 1,c 2,c 3𝒞((c 1c 2)c 3)((X(c 1)Y(c 2))Z(c 3)) α X,Y,Z Day(c) c 1,c 2,c 3𝒞(α c 1,c 2,c 3 𝒞,c)α X(c 1),X(c 2),X(c 3) Top cg */ (X Day(Y DayZ))(c) c 1,c 2,c 3𝒞(c 1(c 2c 3),c)(X(c 1)(Y(c 2)Z(c 3))). \array{ ((X \otimes_{Day} Y) \otimes_{Day} Z)(c) &\simeq& \overset{c_1,c_2, c_3}{\int} \mathcal{C}((c_1 \otimes c_2) \otimes c_3) \wedge ((X(c_1) \wedge Y(c_2)) \wedge Z(c_3)) \\ {}^{\mathllap{ \alpha^{Day}_{X,Y,Z}(c) }}\downarrow && \downarrow^{\mathrlap{ \overset{c_1,c_2,c_3}{\int} \mathcal{C}( \alpha^{\mathcal{C}}_{c_1,c_2,c_3} , c ) \wedge \alpha^{Top^{\ast/}_{cg}}_{X(c_1), X(c_2), X(c_3)} }} \\ (X \otimes_{Day} (Y \otimes_{Day} Z) )(c) &\simeq& \overset{c_1, c_2, c_3}{\int} \mathcal{C}(c_1\otimes ( c_2 \otimes c_3), c ) \wedge (X(c_1) \wedge (Y(c_2) \wedge Z(c_3))) } \,.

It is clear that this satisfies the [[pentagon identity]], since τ 𝒞\tau^{\mathcal{C}} and τ Top cg */\tau^{Top^{\ast/}_{cg}} do.

To see that y(1)y(1) is the tensor unit for Day\otimes_{Day}, use the [[Fubini theorem]] for [[coends]] (prop. ) and then twice the [[co-Yoneda lemma]] (prop. ) to get for any X[𝒞,Top cg */]X \in [\mathcal{C},Top^{\ast/}_{cg}] that

X Dayy(1) =c 1,c 2𝒞𝒞(c 1 𝒟c 2,)X(c 1)𝒞(1,c 2) c 1𝒞c 2𝒞𝒞(c 1 𝒞c 2,)𝒞(1,c 2)X(c 1) c 1𝒞𝒞(c 1 𝒞1,)X(c 1) c 1𝒞𝒞(c 1,)X(c 1) X() X. \begin{aligned} X \otimes_{Day} y(1) & = \overset{c_1,c_2 \in \mathcal{C}}{\int} \mathcal{C}(c_1\otimes_{\mathcal{D}} c_2,-) \wedge X(c_1) \wedge \mathcal{C}(1,c_2) \\ & \simeq \overset{c_1\in \mathcal{C}}{\int} \overset{c_2 \in \mathcal{C}}{\int} \mathcal{C}(c_1\otimes_{\mathcal{C}} c_2,-) \wedge \mathcal{C}(1,c_2) \wedge X(c_1) \\ & \simeq \overset{c_1\in \mathcal{C}}{\int} \mathcal{C}(c_1 \otimes_{\mathcal{C}} 1, -) \wedge X(c_1) \\ & \simeq \overset{c_1\in \mathcal{C}}{\int} \mathcal{C}(c_1, -) \wedge X(c_1) \\ & \simeq X(-) \\ & \simeq X \end{aligned} \,.

Hence the right [[unitor]] of Day convolution comes from the unitor of 𝒞\mathcal{C} under the integral sign:

(X Dayy(1))(c) c 1𝒞(c 11,c)X(c 1) r X Day(c) c 1𝒞(r c 1 𝒞,c)X(c 1) X(c) c 1𝒞(c 1,c)X(c 1). \array{ (X \otimes_{Day} y(1))(c) &\simeq& \overset{c_1}{\int} \mathcal{C}(c_1 \otimes 1, c) \wedge X(c_1) \\ {}^{\mathllap{r^{Day}_{X}(c) } }\downarrow && \downarrow^{\mathrlap{ \overset{c_1}{\int} \mathcal{C}(r^{\mathcal{C}}_{c_1},c) \wedge X(c_1) }} \\ X(c) &\simeq& \overset{c_1}{\int} \mathcal{C}(c_1,c) \wedge X(c_1) } \,.

Analogously for the left unitor. Hence the triangle identity for Day\otimes_{Day} follows from the triangle identity in 𝒞\mathcal{C} under the integral sign.

Similarly, if 𝒞\mathcal{C} has a [[braiding]] τ 𝒞\tau^{\mathcal{C}}, it induces a braiding τ Day\tau^{Day} under the integral sign:

(X DayY)(c) = c 1,c 2𝒞(c 1c 2,c)X(c 1)Y(c 2) τ X,Y Day(c) c 1,c 2𝒞(τ c 1,c 2 𝒞,c)τ X(c(1)),X(c 2) Top */ (Y DayX)(c) = c 1,c 2𝒞(c 2c 1,c)Y(c 2)X(c 1) \array{ (X \otimes_{Day} Y)(c) & = & \overset{c_1,c_2}{\int} \mathcal{C}(c_1 \otimes c_2, c) \wedge X(c_1) \wedge Y(c_2) \\ {}^{\mathllap{\tau}^{Day}_{X,Y}(c)}\downarrow && \downarrow^{\mathrlap{\overset{c_1,c_2}{\int} \mathcal{C}(\tau^{\mathcal{C}}_{c_1,c_2}, c ) \wedge \tau^{Top^{\ast/}}_{X(c(1)), X(c_2)} }} \\ (Y \otimes_{Day} X)(c) & = & \overset{c_1,c_2}{\int} \mathcal{C}(c_2 \otimes c_1, c) \wedge Y(c_2) \wedge X(c_1) }

and the hexagon identity for τ Day\tau^{Day} follows from that for τ 𝒞\tau^{\mathcal{C}} and τ Top cg */\tau^{Top^{\ast/}_{cg}}

Moreover:

Proposition

For (𝒞,,1)(\mathcal{C}, \otimes ,1 ) a [[small category|small]] pointed [[topologically enriched category|topological]] [[symmetric monoidal category]] (def. ), the [[monoidal category]] with [[Day convolution]] ([𝒞,Top cg */], Day,y(1))([\mathcal{C},Top^{\ast/}_{cg}], \otimes_{Day}, y(1)) from def. is a [[closed monoidal category]] (def. ). Its [[internal hom]] [,] Day[-,-]_{Day} is given by the [[end]] (def. )

[X,Y] Day(c)c 1,c 2Maps(𝒞(cc 1,c 2),Maps(X(c 1),Y(c 2)) *) *. [X,Y]_{Day}(c) \simeq \underset{c_1,c_2}{\int} Maps\left( \mathcal{C}(c \otimes c_1,c_2), \; Maps(X(c_1) , Y(c_2))_\ast \right)_\ast \,.
Proof

Using the [[Fubini theorem]] (def. ) and the [[co-Yoneda lemma]] (def. ) and in view of definition of the [[enriched functor category]], there is the following sequence of [[natural isomorphisms]]:

[𝒞,V](X,[Y,Z] Day) cMaps(X(c),c 1,c 2Maps(𝒞(cc 1,c 2),Maps(Y(c 1),Z(c 2)) *) *) * cc 1,c 2Maps(𝒞(cc 1,c 2)X(c)Y(c 1),Z(c 2)) * c 2Maps(c,c 1𝒞(cc 1,c 2)X(c)Y(c 1),Z(c 2)) * c 2Maps((X DayY)(c 2),Z(c 2)) * [𝒞,V](X DayY,Z). \begin{aligned} [\mathcal{C},V]( X, [Y,Z]_{Day} ) & \simeq \underset{c}{\int} Maps\left( X(c), \underset{c_1,c_2}{\int} Maps\left( \mathcal{C}(c \otimes c_1 , c_2), Maps(Y(c_1), Z(c_2))_\ast \right)_\ast \right)_\ast \\ & \simeq \underset{c}{\int} \underset{c_1,c_2}{\int} Maps\left( \mathcal{C}(c \otimes c_1, c_2) \wedge X(c) \wedge Y(c_1) ,\; Z(c_2) \right)_\ast \\ & \simeq \underset{c_2}{\int} Maps\left( \overset{c,c_1}{\int} \mathcal{C}(c \otimes c_1, c_2) \wedge X(c) \wedge Y(c_1) ,\; Z(c_2) \right)_\ast \\ &\simeq \underset{c_2}{\int} Maps\left( (X \otimes_{Day} Y)(c_2), Z(c_2) \right)_\ast \\ &\simeq [\mathcal{C},V](X \otimes_{Day} Y, Z) \end{aligned} \,.
Proposition

In the situation of def. , the [[Yoneda embedding]] c𝒞(c,)c\mapsto \mathcal{C}(c,-) constitutes a [[strong monoidal functor]] (def. )

(𝒞,,1)([𝒞,V], Day,y(1)). (\mathcal{C},\otimes, 1) \hookrightarrow ([\mathcal{C},V], \otimes_{Day}, y(1)) \,.
Proof

That the [[tensor unit]] is respected is part of prop. . To see that the [[tensor product]] is respected, apply the [[co-Yoneda lemma]] (prop. ) twice to get the following natural isomorphism

(y(c 1) Dayy(c 2))(c) d 1,d 2𝒞(d 1d 2,c)𝒞(c 1,d 1)𝒞(c 2,d 2) 𝒞(c 1c 2,c) =y(c 1c 2)(c). \begin{aligned} (y(c_1) \otimes_{Day} y(c_2))(c) & \simeq \overset{d_1, d_2}{\int} \mathcal{C}(d_1 \otimes d_2, c ) \wedge \mathcal{C}(c_1,d_1) \wedge \mathcal{C}(c_2,d_2) \\ & \simeq \mathcal{C}(c_1\otimes c_2 , c ) \\ & = y(c_1 \otimes c_2 )(c) \end{aligned} \,.
Functors with smash product

Since the [[symmetric monoidal smash product of spectra]] discussed below is an instance of [[Day convolution]] (def. ), and since [[ring spectra]] are going to be the [[monoids]] (def. ) with respect to this tensor product, we are interested in characterizing the [[monoid in a monoidal category|monoids]] with respect to Day convolution. These turn out to have a particularly transparent expression as what is called [[functors with smash product]], namely [[lax monoidal functors]] from the base monoidal category to Top cg */Top^{\ast/}_{cg}. Their components are pairing maps of the form

R n 1R n 2R n 1+n 2 R_{n_1} \wedge R_{n_2} \longrightarrow R_{n_1 + n_2}

satisfying suitable conditions. This is the form in which the structure of [[ring spectra]] usually appears in examples. It is directly analogous to how a [[dg-algebra]], which is equivalently a monoid with respect to the [[tensor product of chain complexes]] (example ), is given in components .

Here we introduce the concepts of monoidal functors and of [[functors with smash product]] and prove that they are equivalently the monoids with respect to Day convolution.

Definition

Let (𝒞, 𝒞,1 𝒞)(\mathcal{C},\otimes_{\mathcal{C}}, 1_{\mathcal{C}}) and (𝒟, 𝒟,1 𝒟)(\mathcal{D},\otimes_{\mathcal{D}}, 1_{\mathcal{D}} ) be two (pointed) [[topologically enriched category|topologically enriched]] [[monoidal categories]] (def. ). A topologically enriched lax monoidal functor between them is

  1. a [[topologically enriched functor]]

    F:𝒞𝒟, F \;\colon\; \mathcal{C} \longrightarrow \mathcal{D} \,,
  2. a morphism

    ϵ:1 𝒟F(1 𝒞) \epsilon \;\colon\; 1_{\mathcal{D}} \longrightarrow F(1_{\mathcal{C}})
  3. a [[natural transformation]]

    μ x,y:F(x) 𝒟F(y)F(x 𝒞y) \mu_{x,y} \;\colon\; F(x) \otimes_{\mathcal{D}} F(y) \longrightarrow F(x \otimes_{\mathcal{C}} y)

    for all x,y𝒞x,y \in \mathcal{C}

satisfying the following conditions:

  1. ([[associativity]]) For all objects x,y,z𝒞x,y,z \in \mathcal{C} the following [[commuting diagram|diagram commutes]]

    (F(x) 𝒟F(y)) 𝒟F(z) a F(x),F(y),F(z) 𝒟 F(x) 𝒟(F(y) 𝒟F(z)) μ x,yid idμ y,z F(x 𝒞y) 𝒟F(z) F(x) 𝒟(F(x 𝒞y)) μ x 𝒞y,z μ x,y 𝒞z F((x 𝒞y) 𝒞z) F(a x,y,z 𝒞) F(x 𝒞(y 𝒞z)), \array{ (F(x) \otimes_{\mathcal{D}} F(y)) \otimes_{\mathcal{D}} F(z) &\underoverset{\simeq}{a^{\mathcal{D}}_{F(x),F(y),F(z)}}{\longrightarrow}& F(x) \otimes_{\mathcal{D}}( F(y)\otimes_{\mathcal{D}} F(z) ) \\ {}^{\mathllap{\mu_{x,y} \otimes id}}\downarrow && \downarrow^{\mathrlap{id\otimes \mu_{y,z}}} \\ F(x \otimes_{\mathcal{C}} y) \otimes_{\mathcal{D}} F(z) && F(x) \otimes_{\mathcal{D}} ( F(x \otimes_{\mathcal{C}} y) ) \\ {}^{\mathllap{\mu_{x \otimes_{\mathcal{C}} y , z} } }\downarrow && \downarrow^{\mathrlap{\mu_{ x, y \otimes_{\mathcal{C}} z }}} \\ F( ( x \otimes_{\mathcal{C}} y ) \otimes_{\mathcal{C}} z ) &\underset{F(a^{\mathcal{C}}_{x,y,z})}{\longrightarrow}& F( x \otimes_{\mathcal{C}} ( y \otimes_{\mathcal{C}} z ) ) } \,,

    where a 𝒞a^{\mathcal{C}} and a 𝒟a^{\mathcal{D}} denote the [[associators]] of the monoidal categories;

  2. ([[unitality]]) For all x𝒞x \in \mathcal{C} the following [[commuting diagram|diagrams commutes]]

    1 𝒟 𝒟F(x) ϵid F(1 𝒞) 𝒟F(x) F(x) 𝒟 μ 1 𝒞,x F(x) F( x 𝒞) F(1 𝒞x) \array{ 1_{\mathcal{D}} \otimes_{\mathcal{D}} F(x) &\overset{\epsilon \otimes id}{\longrightarrow}& F(1_{\mathcal{C}}) \otimes_{\mathcal{D}} F(x) \\ {}^{\mathllap{\ell^{\mathcal{D}}_{F(x)}}}\downarrow && \downarrow^{\mathrlap{\mu_{1_{\mathcal{C}}, x }}} \\ F(x) &\overset{F(\ell^{\mathcal{C}}_x )}{\longleftarrow}& F(1 \otimes_{\mathcal{C}} x ) }

    and

    F(x) 𝒟1 𝒟 idϵ F(x) 𝒟F(1 𝒞) r F(x) 𝒟 μ x,1 𝒞 F(x) F(r x 𝒞) F(x 𝒞1), \array{ F(x) \otimes_{\mathcal{D}} 1_{\mathcal{D}} &\overset{id \otimes \epsilon }{\longrightarrow}& F(x) \otimes_{\mathcal{D}} F(1_{\mathcal{C}}) \\ {}^{\mathllap{r^{\mathcal{D}}_{F(x)}}}\downarrow && \downarrow^{\mathrlap{\mu_{x, 1_{\mathcal{C}} }}} \\ F(x) &\overset{F(r^{\mathcal{C}}_x )}{\longleftarrow}& F(x \otimes_{\mathcal{C}} 1 ) } \,,

    where 𝒞\ell^{\mathcal{C}}, 𝒟\ell^{\mathcal{D}}, r 𝒞r^{\mathcal{C}}, r 𝒟r^{\mathcal{D}} denote the left and right [[unitors]] of the two monoidal categories, respectively.

If ϵ\epsilon and alll μ x,y\mu_{x,y} are [[isomorphisms]], then FF is called a strong monoidal functor.

If moreover (𝒞, 𝒞,1 𝒞)(\mathcal{C},\otimes_{\mathcal{C}}, 1_{\mathcal{C}}) and (𝒟, 𝒟,1 𝒟)(\mathcal{D},\otimes_{\mathcal{D}}, 1_{\mathcal{D}} ) are equipped with the structure of [[braided monoidal categories]] (def. ) with [[braidings]] τ 𝒞\tau^{\mathcal{C}} and τ 𝒟\tau^{\mathcal{D}}, respectively, then the lax monoidal functor FF is called a [[braided monoidal functor]] if in addition the following [[commuting diagram|diagram commutes]] for all objects x,y𝒞x,y \in \mathcal{C}

F(x) 𝒞F(y) τ F(x),F(y) 𝒟 F(y) 𝒟F(x) μ x,y μ y,x F(x 𝒞y) F(τ x,y 𝒞) F(y 𝒞x). \array{ F(x) \otimes_{\mathcal{C}} F(y) &\overset{\tau^{\mathcal{D}}_{F(x), F(y)}}{\longrightarrow}& F(y) \otimes_{\mathcal{D}} F(x) \\ {}^{\mathllap{\mu_{x,y}}}\downarrow && \downarrow^{\mathrlap{\mu_{y,x}}} \\ F(x \otimes_{\mathcal{C}} y ) &\underset{F(\tau^{\mathcal{C}}_{x,y} )}{\longrightarrow}& F( y \otimes_{\mathcal{C}} x ) } \,.

A [[homomorphism]] f:(F 1,μ 1,ϵ 1)(F 2,μ 2,ϵ 2)f\;\colon\; (F_1,\mu_1, \epsilon_1) \longrightarrow (F_2, \mu_2, \epsilon_2) between two (braided) lax monoidal functors is a [[monoidal natural transformation]], in that it is a [[natural transformation]] f x:F 1(x)F 2(x)f_x \;\colon\; F_1(x) \longrightarrow F_2(x) of the underlying functors

compatible with the product and the unit in that the following [[commuting diagram|diagrams commute]] for all objects x,y𝒞x,y \in \mathcal{C}:

F 1(x) 𝒟F 1(y) f(x) 𝒟f(y) F 2(x) 𝒟F 2(y) (μ 1) x,y (μ 2) x,y F 1(x 𝒞y) f(x 𝒞y) F 2(x 𝒞y) \array{ F_1(x) \otimes_{\mathcal{D}} F_1(y) &\overset{f(x)\otimes_{\mathcal{D}} f(y)}{\longrightarrow}& F_2(x) \otimes_{\mathcal{D}} F_2(y) \\ {}^{\mathllap{(\mu_1)_{x,y}}}\downarrow && \downarrow^{\mathrlap{(\mu_2)_{x,y}}} \\ F_1(x\otimes_{\mathcal{C}} y) &\underset{f(x \otimes_{\mathcal{C}} y ) }{\longrightarrow}& F_2(x \otimes_{\mathcal{C}} y) }

and

1 𝒟 ϵ 1 ϵ 2 F 1(1 𝒞) f(1 𝒞) F 2(1 𝒞). \array{ && 1_{\mathcal{D}} \\ & {}^{\mathllap{\epsilon_1}}\swarrow && \searrow^{\mathrlap{\epsilon_2}} \\ F_1(1_{\mathcal{C}}) &&\underset{f(1_{\mathcal{C}})}{\longrightarrow}&& F_2(1_{\mathcal{C}}) } \,.

We write MonFun(𝒞,𝒟)MonFun(\mathcal{C},\mathcal{D}) for the resulting [[category]] of lax monoidal functors between monoidal categories 𝒞\mathcal{C} and 𝒟\mathcal{D}, similarly BraidMonFun(𝒞,𝒟)BraidMonFun(\mathcal{C},\mathcal{D}) for the category of braided monoidal functors between [[braided monoidal categories]], and SymMonFun(𝒞,𝒟)SymMonFun(\mathcal{C},\mathcal{D}) for the category of braided monoidal functors between [[symmetric monoidal categories]].

Remark

In the literature the term “monoidal functor” often refers by default to what in def. is called a strong monoidal functor. But for the purpose of the discussion of [[functors with smash product]] below, it is crucial to admit the generality of lax monoidal functors.

If (𝒞, 𝒞,1 𝒞)(\mathcal{C},\otimes_{\mathcal{C}}, 1_{\mathcal{C}}) and (𝒟, 𝒟,1 𝒟)(\mathcal{D},\otimes_{\mathcal{D}}, 1_{\mathcal{D}} ) are [[symmetric monoidal categories]] (def. ) then a [[braided monoidal functor]] (def. ) between them is often called a [[symmetric monoidal functor]].

Proposition

For 𝒞F𝒟G\mathcal{C} \overset{F}{\longrightarrow} \mathcal{D} \overset{G}{\longrightarrow} \mathcal{E} two composable [[lax monoidal functors]] (def. ) between [[monoidal categories]], then their composite FGF \circ G becomes a lax monoidal functor with structure morphisms

ϵ GF:1 ϵ GG(1 𝒟)G(ϵ F)G(F(1 𝒞)) \epsilon^{G\circ F} \;\colon\; 1_{\mathcal{E}} \overset{\epsilon^G}{\longrightarrow} G(1_{\mathcal{D}}) \overset{G(\epsilon^F)}{\longrightarrow} G(F(1_{\mathcal{C}}))

and

μ c 1,c 2 GF:G(F(c 1)) G(F(c 2))μ F(c 1),F(c 2) GG(F(c 1) 𝒟F(c 2))G(μ c 1,c 2 F)G(F(c 1 𝒞c 2)). \mu^{G \circ F}_{c_1,c_2} \;\colon\; G(F(c_1)) \otimes_{\mathcal{E}} G(F(c_2)) \overset{\mu^{G}_{F(c_1), F(c_2)}}{\longrightarrow} G( F(c_1) \otimes_{\mathcal{D}} F(c_2) ) \overset{G(\mu^F_{c_1,c_2})}{\longrightarrow} G(F( c_1 \otimes_{\mathcal{C}} c_2 )) \,.
Proposition

Let (𝒞, 𝒞,1 𝒞)(\mathcal{C},\otimes_{\mathcal{C}}, 1_{\mathcal{C}}) and (𝒟, 𝒟,1 𝒟)(\mathcal{D}, \otimes_{\mathcal{D}},1_{\mathcal{D}}) be two [[monoidal categories]] (def. ) and let F:𝒞𝒟F \;\colon\; \mathcal{C} \longrightarrow \mathcal{D} be a [[lax monoidal functor]] (def. ) between them.

Then for (A,μ A,e A)(A,\mu_A,e_A) a [[monoid in a monoidal category|monoid in]] 𝒞\mathcal{C} (def. ), its image F(A)𝒟F(A) \in \mathcal{D} becomes a monoid (F(A),μ F(A),e F(A))(F(A), \mu_{F(A)}, e_{F(A)}) by setting

μ F(A):F(A) 𝒞F(A)F(A 𝒞A)F(μ A)F(A) \mu_{F(A)} \;\colon\; F(A) \otimes_{\mathcal{C}} F(A) \overset{}{\longrightarrow} F(A \otimes_{\mathcal{C}} A) \overset{F(\mu_A)}{\longrightarrow} F(A)

(where the first morphism is the structure morphism of FF) and setting

e F(A):1 𝒟F(1 𝒞)F(e A)F(A) e_{F(A)} \;\colon\; 1_{\mathcal{D}} \longrightarrow F(1_{\mathcal{C}}) \overset{F(e_A)}{\longrightarrow} F(A)

(where again the first morphism is the corresponding structure morphism of FF).

This construction extends to a functor

Mon(F):Mon(𝒞, 𝒞,1 𝒞)Mon(𝒟, 𝒟,1 𝒟) Mon(F) \;\colon\; Mon(\mathcal{C}, \otimes_{\mathcal{C}}, 1_{\mathcal{C}}) \longrightarrow Mon(\mathcal{D},\otimes_{\mathcal{D}}, 1_{\mathcal{D}})

from the [[category of monoids]] of 𝒞\mathcal{C} (def. ) to that of 𝒟\mathcal{D}.

Moreover, if 𝒞\mathcal{C} and 𝒟\mathcal{D} are [[symmetric monoidal categories]] (def. ) and FF is a [[braided monoidal functor]] (def. ) and AA is a [[commutative monoid]] (def. ) then so is F(A)F(A), and this construction extends to a functor

CMon(F):CMon(𝒞, 𝒞,1 𝒞)CMon(𝒟, 𝒟,1 𝒟). CMon(F) \;\colon\; CMon(\mathcal{C}, \otimes_{\mathcal{C}}, 1_{\mathcal{C}}) \longrightarrow CMon(\mathcal{D},\otimes_{\mathcal{D}}, 1_{\mathcal{D}}) \,.
Proof

This follows immediately from combining the associativity and unitality (and symmetry) constraints of FF with those of AA.

Definition

Let (𝒞, 𝒞,1 𝒞)(\mathcal{C},\otimes_{\mathcal{C}}, 1_{\mathcal{C}}) and (𝒟, 𝒟,1 𝒟)(\mathcal{D},\otimes_{\mathcal{D}}, 1_{\mathcal{D}} ) be two (pointed) [[topologically enriched category|topologically enriched]] [[monoidal categories]] (def. ), and let F:𝒞𝒟F \;\colon\; \mathcal{C} \longrightarrow \mathcal{D} be a [[topologically enriched functor|topologically enriched]] [[lax monoidal functor]] between them, with product operation μ\mu.

Then a left [[module over a monoidal functor|module over the lax monoidal functor]] is

  1. a [[topologically enriched functor]]

    G:𝒞𝒟; G \;\colon\; \mathcal{C} \longrightarrow \mathcal{D} \,;
  2. a [[natural transformation]]

    ρ x,y:F(x) 𝒟G(y)G(x 𝒞y) \rho_{x,y} \;\colon\; F(x) \otimes_{\mathcal{D}} G(y) \longrightarrow G(x \otimes_{\mathcal{C}} y )

such that

  • (action property) For all objects x,y,z𝒞x,y,z \in \mathcal{C} the following [[commuting diagram|diagram commutes]]

    (F(x) 𝒟F(y)) 𝒟G(z) a F(x),F(y),F(z) 𝒟 F(x) 𝒟(F(y) 𝒟G(z)) μ x,yid idρ y,z F(x 𝒞y) 𝒟G(z) F(x) 𝒟(G(x 𝒞y)) ρ x 𝒞y,z ρ x,y 𝒞z G((x 𝒞y) 𝒞z) F(a x,y,z 𝒞) G(x 𝒞(y 𝒞z)), \array{ (F(x) \otimes_{\mathcal{D}} F(y)) \otimes_{\mathcal{D}} G(z) &\underoverset{\simeq}{a^{\mathcal{D}}_{F(x),F(y),F(z)}}{\longrightarrow}& F(x) \otimes_{\mathcal{D}}( F(y)\otimes_{\mathcal{D}} G(z) ) \\ {}^{\mathllap{\mu_{x,y} \otimes id}}\downarrow && \downarrow^{\mathrlap{id\otimes \rho_{y,z}}} \\ F(x \otimes_{\mathcal{C}} y) \otimes_{\mathcal{D}} G(z) && F(x) \otimes_{\mathcal{D}} ( G(x \otimes_{\mathcal{C}} y) ) \\ {}^{\mathllap{\rho_{x \otimes_{\mathcal{C}} y , z} } }\downarrow && \downarrow^{\mathrlap{\rho_{ x, y \otimes_{\mathcal{C}} z }}} \\ G( ( x \otimes_{\mathcal{C}} y ) \otimes_{\mathcal{C}} z ) &\underset{F(a^{\mathcal{C}}_{x,y,z})}{\longrightarrow}& G( x \otimes_{\mathcal{C}} ( y \otimes_{\mathcal{C}} z ) ) } \,,

A [[homomorphism]] f:(G 1,ρ 1)(G 2,ρ 2)f\;\colon\; (G_1, \rho_1) \longrightarrow (G_2,\rho_2) between two modules over a monoidal functor (F,μ,ϵ)(F,\mu,\epsilon) is

  • a [[natural transformation]] f x:G 1(x)G 2(x)f_x \;\colon\; G_1(x) \longrightarrow G_2(x) of the underlying functors

compatible with the action in that the following [[commuting diagram|diagram commutes]] for all objects x,y𝒞x,y \in \mathcal{C}:

F(x) 𝒟G 1(y) id 𝒟f(y) F(x) 𝒟G 2(y) (ρ 1) x,y (rhi 2) x,y G 1(x 𝒞y) f(x 𝒞y) G 2(x 𝒞y) \array{ F(x) \otimes_{\mathcal{D}} G_1(y) &\overset{id \otimes_{\mathcal{D}} f(y)}{\longrightarrow}& F(x) \otimes_{\mathcal{D}} G_2(y) \\ {}^{\mathllap{(\rho_1)_{x,y}}}\downarrow && \downarrow^{\mathrlap{(\rhi_2)_{x,y}}} \\ G_1(x\otimes_{\mathcal{C}} y) &\underset{f(x \otimes_{\mathcal{C}} y ) }{\longrightarrow}& G_2(x \otimes_{\mathcal{C}} y) }

We write FModF Mod for the resulting category of modules over the monoidal functor FF.

Now we may finally state the main proposition on [[functors with smash product]]:

Proposition

Let (𝒞,,1)(\mathcal{C},\otimes, 1) be a pointed [[topologically enriched category|topologically enriched]] ([[symmetric monoidal category|symmetric]]) [[monoidal category]] (def. ). Regard (Top cg */,,S 0)(Top_{cg}^{\ast/}, \wedge , S^0) as a topological [[symmetric monoidal category]] as in example .

Then ([[commutative monoid in a symmetric monoidal category|commutative]]) [[monoid in a monoidal category|monoids in]] (def. ) the [[Day convolution]] monoidal category ([𝒞,Top cg */], Day,y(1 𝒞))([\mathcal{C}, Top^{\ast/}_{cg}], \otimes_{Day}, y(1_{\mathcal{C}})) of prop. are equivalent to ([[braided monoidal functor|braided]]) [[lax monoidal functors]] (def. ) of the form

(𝒞,,1)(Top cg *,,S 0), (\mathcal{C},\otimes, 1) \longrightarrow (Top^{\ast}_{cg}, \wedge, S^0) \,,

called [[functors with smash products]] on 𝒞\mathcal{C}, i.e. there are [[equivalences of categories]] of the form

Mon([𝒞,Top cg */], Day,y(1 𝒞)) MonFunc(𝒞,Top cg */) CMon([𝒞,Top cg */], Day,y(1 𝒞)) SymMonFunc(𝒞,Top cg */). \begin{aligned} Mon([\mathcal{C},Top^{\ast/}_{cg}], \otimes_{Day}, y(1_{\mathcal{C}})) &\simeq MonFunc(\mathcal{C},Top^{\ast/}_{cg}) \\ CMon([\mathcal{C},Top^{\ast/}_{cg}], \otimes_{Day}, y(1_{\mathcal{C}})) &\simeq SymMonFunc(\mathcal{C},Top^{\ast/}_{cg}) \end{aligned} \,.

Moreover, [[module objects]] over these monoid objects are equivalent to the corresponding [[modules over monoidal functors]] (def. ).

This is stated in some form in (Day 70, example 3.2.2). It is highlighted again in (MMSS 00, prop. 22.1).

Proof

By definition , a [[lax monoidal functor]] F:𝒞Top cg */F \colon \mathcal{C} \to Top^{\ast/}_{cg} is a topologically enriched functor equipped with a morphism of [[pointed topological spaces]] of the form

S 0F(1 𝒞) S^0 \longrightarrow F(1_{\mathcal{C}})

and equipped with a [[natural transformation|natural]] system of maps of pointed topological spaces of the form

F(c 1)F(c 2)F(c 1 𝒞c 2) F(c_1) \wedge F(c_2) \longrightarrow F(c_1 \otimes_{\mathcal{C}} c_2)

for all c 1,c 2𝒞c_1,c_2 \in \mathcal{C}.

Under the [[Yoneda lemma]] (prop. ) the first of these is equivalently a morphism in [𝒞,Top cg */][\mathcal{C}, Top^{\ast/}_{cg}] of the form

y(S 0)F. y(S^0) \longrightarrow F \,.

Moreover, under the [[natural isomorphism]] of corollary the second of these is equivalently a morphism in [𝒞,Top cg */][\mathcal{C}, Top^{\ast/}_{cg}] of the form

F DayFF. F \otimes_{Day} F \longrightarrow F \,.

Translating the conditions of def. satisfied by a [[lax monoidal functor]] through these identifications gives precisely the conditions of def. on a ([[commutative monoid in a symmetric monoidal category|commutative]]) [[monoid in a monoidal category|monoid in]] object FF under Day\otimes_{Day}.

Similarly for [[module objects]] and [[modules over monoidal functors]].

Proposition

Let f:𝒞𝒟f \;\colon\; \mathcal{C} \longrightarrow \mathcal{D} be a [[lax monoidal functor]] (def. ) between pointed [[topologically enriched category|topologically enriched]] [[monoidal categories]] (def. ). Then the induced functor

f *:[𝒟,Top cg */][𝒞,Top cg *] f^\ast \;\colon\; [\mathcal{D}, Top^{\ast/}_{cg}] \longrightarrow [\mathcal{C}, Top_{cg}^{\ast}]

given by (f *X)(c)X(f(c))(f^\ast X)(c)\coloneqq X(f(c)) preserves [[monoid in a monoidal category|monoids]] under [[Day convolution]]

f *:Mon([𝒟,Top cg */], Day,y(1 𝒟))Mon([𝒞,Top cg *], Day,y(1 𝒞) f^\ast \;\colon\; Mon([\mathcal{D}, Top^{\ast/}_{cg}], \otimes_{Day}, y(1_{\mathcal{D}})) \longrightarrow Mon([\mathcal{C}, Top_{cg}^{\ast}], \otimes_{Day}, y(1_{\mathcal{C}})

Moreover, if 𝒞\mathcal{C} and 𝒟\mathcal{D} are [[symmetric monoidal categories]] (def. ) and ff is a [[braided monoidal functor]] (def. ), then f *f^\ast also preserves [[commutative monoids in a symmetric monoidal category|commutative monoids]]

f *:CMon([𝒟,Top cg */], Day,y(1 𝒟))CMon([𝒞,Top cg *], Day,y(1 𝒞). f^\ast \;\colon\; CMon([\mathcal{D}, Top^{\ast/}_{cg}], \otimes_{Day}, y(1_{\mathcal{D}})) \longrightarrow CMon([\mathcal{C}, Top_{cg}^{\ast}], \otimes_{Day}, y(1_{\mathcal{C}}) \,.

Similarly, for

AMon([𝒟,Top cg */], Day,y(1 𝒟)) A \in Mon([\mathcal{D}, Top^{\ast/}_{cg}], \otimes_{Day}, y(1_{\mathcal{D}}))

any fixed monoid, then f *f^\ast sends AA-[[module object|modules]] to f *(A)f^\ast(A)-modules

f *:AMod(𝒟)(f *A)Mod(𝒞). f^\ast \;\colon\; A Mod(\mathcal{D}) \longrightarrow (f^\ast A)Mod(\mathcal{C}) \,.
Proof

This is an immediate corollary of prop. , since the composite of two (braided) lax monoidal functors is itself canonically a (braided) lax monoidal functor by prop. .

𝕊\mathbb{S}-Modules

We give a unified discussion of the categories of

  1. [[sequential spectra]]

  2. [[symmetric spectra]]

  3. [[orthogonal spectra]]

  4. pre-[[excisive functors]]

(all in [[topological spaces]]) as [[categories of modules]] with respect to [[Day convolution]] monoidal structures on [[Top]]-[[enriched functor categories]] over restrictions to [[faithful functor|faithful]] sub-[[sites]] of the canonical representative of the [[sphere spectrum]] as a pre-excisive functor on Top fin */Top^{\ast/}_{fin}.

This approach is due to (Mandell-May-Schwede-Shipley 00) following (Hovey-Shipley-Smith 00).

Pre-Excisive functors

We consider an almost tautological construction of a pointed topologically enriched category equipped with a closed symmetric monoidal product: the category of [[pre-excisive functors]]. Then we show that this tautological category restricts, in a certain sense, to the category of [[sequential spectra]]. However, under this restriction the symmetric monoidal product breaks, witnessing the lack of a functorial [[smash product of spectra]] on sequential spectra. However from inspection of this failure we see that there are categories of [[structured spectra]] “in between” those of all pre-excisive functors and plain sequential spectra, notably the categories of [[orthogonal spectra]] and of [[symmetric spectra]]. These intermediate categories retain the concrete tractable nature of sequential spectra, but are rich enough to also retain the symmetric monoidal product inherited from pre-excisive functors: this is the [[symmetric monoidal smash product of spectra]] that we are after.

Literature (MMSS 00, Part I and Part III)

\,

Definition

Write

ι fin:Top cg,fin */Top cg */ \iota_{fin}\;\colon\; Top^{\ast/}_{cg,fin} \hookrightarrow Top^{\ast/}_{cg}

for the [[full subcategory]] of [[pointed topological spaces|pointed]] [[compactly generated topological spaces]] (def.) on those that admit the structure of a [[finite CW-complex]] (a [[CW-complex]] (def.) with a [[finite number]] of cells).

We say that the pointed topological [[enriched functor category]] (def. )

Exc(Top cg)[Top cg,fin */,Top cg */] Exc(Top_{cg}) \coloneqq [Top^{\ast/}_{cg,fin}, Top^{\ast/}_{cg}]

is the category of [[pre-excisive functors]]. (We had previewed this in [[Introduction to Stable homotopy theory – P|Part P]], this example).

Write

𝕊 excy(S 0)Top cg,fin */(S 0,) \mathbb{S}_{exc} \coloneqq y(S^0) \coloneqq Top^{\ast/}_{cg,fin}(S^0,-)

for the [[representable functor|functor co-represented]] by [[0-sphere]]. This is equivalently the inclusion ι fin\iota_{fin} itself:

𝕊 exc=ι fin:KK. \mathbb{S}_{exc} = \iota_{fin} \;\colon\; K \mapsto K \,.

We call this the standard incarnation of the [[sphere spectrum]] as a pre-excisive functor.

By prop. the [[smash product]] of [[pointed topological spaces|pointed]] [[compactly generated topological spaces]] induces the structure of a [[closed monoidal category|closed]] (def. ) [[symmetric monoidal category]] (def. )

(Exc(Top cg), Day,𝕊 exc) \left( Exc(Top_{cg}) ,\; \wedge \coloneqq \otimes_{Day} ,\; \mathbb{S}_{exc} \right)

with

  1. [[tensor unit]] the [[sphere spectrum]] 𝕊 exc\mathbb{S}_{exc};

  2. [[tensor product]] the [[Day convolution product]] Day\otimes_{Day} from def. ,

    called the [[symmetric monoidal smash product of spectra]] for the model of pre-excisive functors;

  3. [[internal hom]] the dual operation [,] Day[-,-]_{Day} from prop. ,

    called the [[mapping spectrum]] construction for pre-excisive functors.

Remark

By example the [[sphere spectrum]] incarnated as a pre-excisive functor 𝕊 exc\mathbb{S}_{exc} (according to def. ) is canonically a [[commutative monoid in a symmetric monoidal category|commutative monoid in]] the category of pre-excisive functors (def. ).

Moreover, by example , every object of Exc(Top cg)Exc(Top_{cg}) (def. ) is canonically a [[module object]] over 𝕊 exc\mathbb{S}_{exc}. We may therefore tautologically identify the category of pre-excisive functors with the [[module category]] over the sphere spectrum:

Exc(Top cg)𝕊 excMod. Exc(Top_{cg}) \simeq \mathbb{S}_{exc}Mod \,.
Lemma

Identified as a [[functor with smash product]] under prop. , the pre-excisive [[sphere spectrum]] 𝕊 exc\mathbb{S}_{exc} from def. is given by the identity natural transformation

μ (K 1,K 2):𝕊 exc(K 1)𝕊 exc(K 2)=K 1K 2=K 1K 2=𝕊 exc(K 1K 2). \mu_{(K_1,K_2)} \;\colon\; \mathbb{S}_{exc}(K_1) \wedge \mathbb{S}_{exc}(K_2) = K_1 \wedge K_2 \overset{=}{\longrightarrow} K_1 \wedge K_2 = \mathbb{S}_{exc}(K_1 \wedge K_2) \,.
Proof

We claim that this is in fact the unique structure of a [[monoidal functor]] that may be imposed on the canonical inclusion ι:Top cg,fin */Top cg */\iota \;\colon\; Top^{\ast/}_{cg,fin} \hookrightarrow Top^{\ast/}_{cg}, hence it must be the one in question. To see the uniqueness, observe that naturality of the matural transformation μ\mu in particular says that there are commuting squares of the form

S 0S 0 = S 0S 0 x 1,x 2 x 1,x 2 K 1K 2 μ K 1,K 2 K 1K 2, \array{ S^0 \wedge S^0 &\overset{=}{\longrightarrow}& S^0 \wedge S^0 \\ {}^{\mathllap{x_1,x_2}}\downarrow && \downarrow^{\mathrlap{x_1,x_2}} \\ K_1 \wedge K_2 &\underset{\mu_{K_1, K_2}}{\longrightarrow}& K_1 \wedge K_2 } \,,

where the vertical morphisms pick any two points in K 1K_1 and K 2K_2, respectively, and where the top morphism is necessarily the canonical identification since there is only one single isomorphism S 0S 0S^0 \to S^0, namely the identity. This shows that the bottom horizontal morphism has to be the identity on all points, hence has to be the identity.

We now consider restricting the domain of the pre-excisive functors of def. .

Definition

Define the following [[pointed topologically enriched categories|pointed topologically enriched]] (def.) [[symmetric monoidal categories]] (def. ):

  1. SeqSeq is the category whose objects are the [[natural numbers]] and which has only identity morphisms and [[zero morphisms]] on these objects, hence the [[hom-spaces]] are

    Seq(n 1,n 2){S 0 forn 1=n 2 * otherwise Seq(n_1,n_2) \;\coloneqq\; \left\{ \array{ S^0 & for\; n_1 = n_2 \\ \ast & otherwise } \right.

    The tensor product is the addition of natural numbers, =+\otimes = +, and the [[tensor unit]] is 0. The [[braiding]] is, necessarily, the identity.

  2. SymSym is the standard [[skeletal category|skeleton]] of the [[core]] of [[FinSet]] with [[zero morphisms]] adjoined: its [[objects]] are the [[finite sets]] n¯{1,,n}\overline{n} \coloneqq \{1, \cdots,n\} for nn \in \mathbb{N} (hence 0¯\overline{0} is the [[empty set]]), all non-[[zero morphism|zero]] morphisms are [[automorphisms]] and the [[automorphism group]] of {1,,n}\{1,\cdots,n\} is the [[symmetric group]] Σ(n)\Sigma(n) on nn elements, hence the [[hom-spaces]] are the following [[discrete topological spaces]]:

    Sym(n 1,n 2){(Σ(n 1)) + forn 1=n 2 * otherwise Sym(n_1, n_2) \;\coloneqq\; \left\{ \array{ (\Sigma(n_1))_+ & for \; n_1 = n_2 \\ \ast & otherwise } \right.

    The [[tensor product]] is the [[disjoint union]] of sets, tensor unit is the [[empty set]]. The [[braiding]]

    τ n 1,n 2 Sym:n 1¯n 2¯n 2¯n 1¯ \tau^{Sym}_{n_1 , n_2} \;\colon\; \overline{n_1} \cup \overline{n_2} \overset{}{\longrightarrow} \overline{n_2} \cup \overline{n_1}

    is given by the canonical [[permutation]] in Σ(n 1+n 2)\Sigma(n_1+n_2) that [[shuffle|shuffles]] the first n 1n_1 elements past the remaining n 2n_2 elements

    (MMSS 00, example 4.2)

  3. OrthOrth has as objects the finite dimenional real linear [[inner product spaces]] ( n,,)(\mathbb{R}^n, \langle -,-\rangle) and as non-zero morphisms the [[linear map|linear]] [[isometry|isometric]] [[isomorphisms]] between these; hence the [[automorphism group]] of the object ( n,,)(\mathbb{R}^n, \langle -,-\rangle) is the [[orthogonal group]] O(n)O(n); the monoidal product is [[direct sum]] of linear spaces, the tensor unit is the 0-vector space; again we turn this into a Top cg */Top^{\ast/}_{cg}-enriched category by adjoining a basepoint to the hom-spaces;

    Orth(V 1,V 2){O(V 1) + fordim(V 1)=dim(V 2) * otherwise Orth(V_1,V_2) \;\coloneqq\; \left\{ \array{ O(V_1)_+ & for \; dim(V_1) = dim(V_2) \\ \ast & otherwise } \right.

    The [[tensor product]] is the [[direct sum]] of linear inner product spaces, tensor unit is the 0-vector space. The [[braiding]]

    τ V 1,V 2 Orth:V 1V 2V 2V 1 \tau^{Orth}_{V_1,V_2} \;\colon\; V_1 \oplus V_2 \longrightarrow V_2 \oplus V_1

    is the canonical orthogonal transformation that switches the summands.

    (MMSS 00, example 4.4)

Notice that in the notation of example

  1. the [[full subcategory]] of OrthOrth on VV is B(O(V) +)\mathbf{B}(O(V)_+);

  2. the [[full subcategory]] of SymSym on {1,,n}\{1,\cdots,n\} is B(Σ(n) +)\mathbf{B}(\Sigma(n)_+);

  3. the [[full subcategory]] of SeqSeq on nn is B(1 +)\mathbf{B}(1_+).

Moreover, after discarding the [[zero morphisms]], then these categories are the disjoint union of categories of the form BO(n)\mathbf{B}O(n), BΣ(n)\mathbf{B}\Sigma(n) and B1=*\mathbf{B}1 = \ast, respectively.

There is a sequence of canonical [[faithful functor|faithful]] pointed topological [[subcategory]] inclusions

Seq seq Sym sym Orth orth Top cg,fin */ n {1,,n} n S n, \array{ Seq &\stackrel{seq}{\hookrightarrow}& Sym &\stackrel{sym}{\hookrightarrow}& Orth &\stackrel{orth}{\hookrightarrow}& Top_{cg,fin}^{\ast/} \\ n &\mapsto& \{1,\cdots, n\} &\mapsto& \mathbb{R}^n &\mapsto& S^n } \,,

into the pointed topological category of pointed compactly generated topological spaces of finite CW-type (def. ).

Here S VS^V denotes the [[one-point compactification]] of VV. On morphisms sym:(Σ n) +(O(n)) +sym \colon (\Sigma_n)_+ \hookrightarrow (O(n))_+ is the canonical inclusion of [[permutation]] matrices into [[orthogonal group|orthogonal]] matrices and orth:O(V) +Aut(S V)orth \colon O(V)_+ \hookrightarrow Aut(S^V) is on O(V)O(V) the [[topological subspace]] inclusions of the pointed [[homeomorphisms]] S VS VS^V \to S^V that are induced under forming [[one-point compactification]] from linear isometries of VV (“[[representation spheres]]”).

Below we will often use these identifications to write just “nn” for any of these objects, leaving implicit the identifications n{1,,n}S nn \mapsto \{1, \cdots, n\} \mapsto S^n.

Consider the pointed topological diagram categries (def. , exmpl.) over these categories:

  • [Seq,Top cg */][Seq,Top^{\ast/}_{cg}] is called the category of sequences of pointed topological spaces (e.g. HSS 00, def. 2.3.1);

  • [Sym,Top cg */][Sym,Top^{\ast/}_{cg}] is called the category of [[symmetric sequences]] (e.g. HSS 00, def. 2.1.1);

  • [Orth,Top cg */][Orth, Top^{\ast/}_{cg}] is called the category of orthogonal sequences.

Consider the sequence of restrictions of topological diagram categories, according to prop. along the above inclusions:

Exc(Top cg)orth *[Orth,Top cg */]sym *[Sym,Top cg */]seq *[Seq,Top cg */]. Exc(Top_{cg}) \overset{orth^\ast}{\longrightarrow} [Orth,Top^{\ast/}_{cg}] \overset{sym^\ast}{\longrightarrow} [Sym,Top^{\ast/}_{cg}] \overset{seq^\ast}{\longrightarrow} [Seq,Top^{\ast/}_{cg}] \,.

Write

𝕊 orthorth *𝕊 exc,𝕊 symsym *𝕊 orth,𝕊 seqseq *𝕊 sym \mathbb{S}_{orth} \coloneqq orth^\ast \mathbb{S}_{exc} \,, \;\;\;\;\;\;\;\; \mathbb{S}_{sym} \coloneqq sym^\ast \mathbb{S}_{orth} \,, \;\;\;\;\;\;\;\; \mathbb{S}_{seq} \coloneqq seq^\ast \mathbb{S}_{sym}

for the restriction of the excisive functor incarnation of the [[sphere spectrum]] (from def. ) along these inclusions.

Proposition

The functors seqseq, symsym and orthorth in def. become [[strong monoidal functors]] (def. ) when equipped with the canonical isomorphisms

seq(n 1)seq(n 2)={1,,n 1}{1,,n 2}{1,,n 1+n 2}=seq(n 1+n 2) seq(n_1) \cup seq(n_2) = \{1,\cdots, n_1\} \cup \{1, \cdots, n_2\} \simeq \{1, \cdots, n_1+ n_2\} = seq(n_1 + n_2)

and

sym({1,,n 1})sym({1,,n 2})= n 1 n 2 n 1+n 2=sym({1,,n 1}{1,,n 2}) sym(\{1,\cdots,n_1\}) \oplus sym(\{1,\cdots,n_2\}) = \mathbb{R}^{n_1} \oplus \mathbb{R}^{n_2} \simeq \mathbb{R}^{n_1 + n_2} = sym(\{1,\cdots, n_1\} \cup \{1,\cdots, n_2\})

and

orth(V 1)orth(V 2)=S V 1S V 2S V 1V 2=orth(V 1V 2). orth(V_1) \wedge orth(V_2) = S^{V_1} \wedge S^{V_2} \simeq S^{V_1 \oplus V_2} = orth(V_1 \oplus V_2) \,.

Moreover, orthorth and symsym are [[braided monoidal functors]] (def. ) (hence [[symmetric monoidal functors]], remark ). But seqseq is not braided monoidal.

Proof

The first statement is clear from inspection.

For the second statement it is sufficient to observe that all the nontrivial braiding of [[n-spheres]] in Top cg */Top^{\ast/}_{cg} is given by the maps induced from exchanging coordinates in the realization of nn-spheres as [[one-point compactifications]] of [[Cartesian spaces]] S n( n) *S^n \simeq (\mathbb{R}^n)^\ast. This corresponds precisely to the action of the [[symmetric group]] inside the [[orthogonal group]] acting via the canonical action of the orthogonal group on n\mathbb{R}^n. This shows that symsym and orthorth are braided, for they include precisely these objects (the nn-spheres) with these braidings on them. Finally it is clear that seqseq is not braided, because the braiding on SeqSeq is trivial, while that on SymSym is not, so seqseq necessrily fails to preserve precisely these non-trivial isomorphisms.

Remark

Since the standard excisive incarnation 𝕊 exc\mathbb{S}_{exc} of the [[sphere spectrum]] (def. ) is the [[tensor unit]] with repect to the [[Day convolution]] product on pre-excisive functors, and since it is therefore canonically a [[commutative monoid]], by example , prop. says that the restricted sphere spectra 𝕊 orth\mathbb{S}_{orth}, 𝕊 sym\mathbb{S}_{sym} and 𝕊 seq\mathbb{S}_{seq} are still [[monoid object|monoids]], and that under restriction every [[pre-excisive functor]], regarded as a 𝕊 exc\mathbb{S}_{exc}-[[module object|module]] via remark , canonically becomes a [[module object|module]] under the restricted sphere spectrum:

orth * :Exc(Top cg)𝕊 excMod𝕊 orthMod sym * :Exc(Top cg)𝕊 excMod𝕊 symMod seq * :Exc(Top cg)𝕊 excMod𝕊 seqMod. \begin{aligned} orth^\ast & \colon\; Exc(Top_{cg}) \simeq \mathbb{S}_{exc} Mod \longrightarrow \mathbb{S}_{orth} Mod \\ sym^\ast &\colon\; Exc(Top_{cg}) \simeq \mathbb{S}_{exc} Mod \longrightarrow \mathbb{S}_{sym} Mod \\ seq^\ast &\colon\; Exc(Top_{cg}) \simeq \mathbb{S}_{exc} Mod \longrightarrow \mathbb{S}_{seq} Mod \end{aligned} \,.

Since all three functors orthorth, symsym and seqseq are strong monoidal functors by prop. , all three restricted sphere spectra 𝕊 orth\mathbb{S}_{orth}, 𝕊 sym\mathbb{S}_{sym} and 𝕊 seq\mathbb{S}_{seq} canonically are [[monoids]], by prop. . Moreover, according to prop. , orthorth and symsym are [[braided monoidal functors]], while functor seqseq is not braided, therefore prop. furthermore gives that 𝕊 orth\mathbb{S}_{orth} and 𝕊 sym\mathbb{S}_{sym} are [[commutative monoid in a symmetric monoidal category|commutative monoids]], while 𝕊 seq\mathbb{S}_{seq} is not commutative:

[[sphere spectrum]]𝕊 exc\mathbb{S}_{exc}𝕊 orth\mathbb{S}_{orth}𝕊 sym\mathbb{S}_{sym}𝕊 seq\mathbb{S}_{seq}
[[monoid in a monoidal categorymonoid]]yesyesyes
[[commutative monoid in a symmetric monoidal categorycommutative monoid]]yesyesyes
[[tensor unit]]yesnonono

Explicitly:

Lemma

The monoids 𝕊 dia\mathbb{S}_{dia} from def. are, when identified as [[functors with smash product]] via prop. given by assigning

𝕊 seq:nS n \mathbb{S}_{seq} \;\colon\; n \mapsto S^{n}
𝕊 sym:n¯S n \mathbb{S}_{sym} \;\colon\; \overline{n} \mapsto S^n
𝕊 orth:VS V, \mathbb{S}_{orth} \;\colon\; V \mapsto S^V \,,

respectively, with product given by the canonical isomorphisms

S V 1S V 2S V 1V 2. S^{V_1} \wedge S^{V_2} \longrightarrow S^{V_1 \oplus V_2} \,.
Proof

By construction these functors with smash products are the composites, according to prop. , of the monoidal functors seqseq, symsym, orthorth, respectively, with the lax monoidal functor corresponding to 𝕊 exc\mathbb{S}_{exc}. The former have as structure maps the canonical identifications by definition, and the latter has as structure map the canonical identifications by lemmma .

Proposition

There is an [[equivalence of categories]]

() seq:𝕊 seqModSeqSpec(Top cg) (-)^{seq} \;\colon\; \mathbb{S}_{seq} Mod \overset{}{\longrightarrow} SeqSpec(Top_{cg})

which identifies the [[category of modules]] (def. ) over the [[monoid object|monoid]] 𝕊 seq\mathbb{S}_{seq} (remark ) in the [[Day convolution]] monoidal structure (prop. ) over the topological functor category [Seq,Top cg */][Seq,Top^{\ast/}_{cg}] from def. with the category of [[sequential spectra]] (def.)

Under this equivalence, an 𝕊 seq\mathbb{S}_{seq}-module XX is taken to the sequential pre-spectrum X seqX^{seq} whose component spaces are the values of the [[pre-excisive functor]] XX on the standard [[n-sphere]] S n=(S 1) nS^n = (S^1)^{\wedge n}

(X seq) nX(seq(n))=X(S n) (X^{seq})_n \coloneqq X(seq(n)) = X(S^n)

and whose structure maps are the images of the action morphisms

𝕊 seq DayXX \mathbb{S}_{seq} \otimes_{Day} X \longrightarrow X

under the isomorphism of corollary

𝕊 seq(n 1)X(n 1)X n 1+n 2 \mathbb{S}_{seq}(n_1) \wedge X(n_1) \longrightarrow X_{n_1 + n_2}

evaluated at n 1=1n_1 = 1

𝕊 seq(1)X(n) X n+1 S 1X n X n+1. \array{ \mathbb{S}_{seq}(1) \wedge X(n) &\longrightarrow& X_{n+1} \\ {}^{\mathllap{\simeq}}\downarrow && \downarrow^{\mathrlap{\simeq}} \\ S^1 \wedge X_n &\longrightarrow& X_{n+1} } \,.

(Hovey-Shipley-Smith 00, prop. 2.3.4)

Proof

After unwinding the definitions, the only point to observe is that due to the action property,

𝕊 seq Day𝕊 seq DayX id Dayρ 𝕊 seq DayX μ Dayid ρ 𝕊 seq DayX ρ X \array{ \mathbb{S}_{seq} \otimes_{Day} \mathbb{S}_{seq} \otimes_{Day} X &\overset{id \otimes_{Day} \rho}{\longrightarrow}& \mathbb{S}_{seq} \otimes_{Day} X \\ {}^{\mathllap{\mu \otimes_{Day} id } }\downarrow && \downarrow^{\mathrlap{\rho}} \\ \mathbb{S}_{seq} \otimes_{Day} X &\underset{\rho}{\longrightarrow}& X }

any 𝕊 seq\mathbb{S}_{seq}-action

ρ:𝕊 seq DayXX \rho \;\colon\; \mathbb{S}_{seq} \otimes_{Day} X \longrightarrow X

is indeed uniquely fixed by the components of the form

𝕊 seq(1)X(n)X(n). \mathbb{S}_{seq}(1) \wedge X(n) \longrightarrow X(n) \,.

This is because under corollary the action property is identified with the componentwise property

S n 1S n 2X n 3 idρ n 2,n 3 S n 1X n 2+n 3 ρ n 1,n 2+n 3 S n 1+n 2X n 3 ρ n 1+n 2,n 3 X n 1+n 2+n 3, \array{ S^{n_1} \wedge S^{n_2} \wedge X_{n_3} &\overset{id \wedge \rho_{n_2,n_3}}{\longrightarrow}& S^{n_1} \wedge X_{n_2 + n_3} \\ {}^{\mathllap{\simeq}}\downarrow && \downarrow^{\mathrlap{\rho_{n_1,n_2+n_3}}} \\ S^{n_1 + n_2} \wedge X_{n_3} &\underset{\rho_{n_1+n_2,n_3}}{\longrightarrow}& X_{n_1 + n_2 + n_3} } \,,

where the left vertical morphism is an isomorphism by the nature of 𝕊 seq\mathbb{S}_{seq}. Hence this fixes the components ρ n,n\rho_{n',n} to be the nn'-fold composition of the structure maps σ nρ(1,n)\sigma_n \coloneqq \rho(1,n).

However, since, by remark , 𝕊 seq\mathbb{S}_{seq} is not commutative, there is no tensor product induced on SeqSpec(Top cg)SeqSpec(Top_{cg}) under the identification in prop. . But since 𝕊 orth\mathbb{S}_{orth} and 𝕊 sym\mathbb{S}_{sym} are commutative monoids by remark , it makes sense to consider the following definition.

Definition

In the terminology of remark we say that

OrthSpec(Top cg)𝕊 orthMod OrthSpec(Top_{cg}) \coloneqq \mathbb{S}_{orth} Mod

is the [[category]] of [[orthogonal spectra]]; and that

SymSpec(Top cg)𝕊 symMod SymSpec(Top_{cg}) \coloneqq \mathbb{S}_{sym} Mod

is the [[category]] of [[symmetric spectra]].

By remark and by prop. these categories canonically carry a [[symmetric monoidal category|symmetric monoidal]] [[tensor product]] 𝕊 orth\otimes_{\mathbb{S}_{orth}} and 𝕊 seq\otimes_{\mathbb{S}_{seq}}, respectively. This we call the [[symmetric monoidal smash product of spectra]]. We usually just write for short

𝕊 orth:OrthSpec(Top cg)×OrthSpec(Top cg)OrthSpec(Top cg) \wedge \coloneqq \otimes_{\mathbb{S}_{orth}} \;\colon\; OrthSpec(Top_{cg}) \times OrthSpec(Top_{cg}) \longrightarrow OrthSpec(Top_{cg})

and

𝕊 sym:SymSpec(Top cg)×SymSpec(Top cg)SymSpec(Top cg) \wedge \coloneqq \otimes_{\mathbb{S}_{sym}} \;\colon\; SymSpec(Top_{cg}) \times SymSpec(Top_{cg}) \longrightarrow SymSpec(Top_{cg})

In the next section we work out what these symmetric monoidal categories of orthogonal and of symmetric spectra look like more explicitly.

Symmetric and orthogonal spectra

We now define [[symmetric spectra]] and [[orthogonal spectra]] and their symmetric monoidal smash product. We proceed by giving the explicit definitions and then checking that these are equivalent to the abstract definition from above.

Literature. ( Hovey-Shipley-Smith 00, section 1, section 2, Schwede 12, chapter I)

\,

Definition

A topological [[symmetric spectrum]] XX is

  1. a sequence {X nTop cg */|n}\{X_n \in Top_{cg}^{\ast/}\;\vert\; n \in \mathbb{N}\} of [[pointed topological space|pointed]] [[compactly generated topological spaces]];

  2. a basepoint preserving continuous right [[action]] of the [[symmetric group]] Σ(n)\Sigma(n) on X nX_n;

  3. a sequence of morphisms σ n:S 1X nX n+1\sigma_n \colon S^1 \wedge X_n \longrightarrow X_{n+1}

such that

  • for all n,kn, k \in \mathbb{N} the [[composition|composite]]

    S kX nS k1S 1X nidσ nS k1X n+1S k2S 1X n+2idσ n+1σ n+k1X n+k S^{k} \wedge X_n \simeq S^{k-1} \wedge S^1 \wedge X_n \stackrel{id \wedge \sigma_n }{\longrightarrow} S^{k-1} \wedge X_{n+1} \simeq S^{k-2}\wedge S^1 \wedge X_{n+2} \stackrel{id \wedge \sigma_{n+1}}{\longrightarrow} \cdots \stackrel{\sigma_{n+k-1}}{\longrightarrow} X_{n+k}

    [[intertwiner|intertwines]] the Σ(n)×Σ(k)\Sigma(n) \times \Sigma(k)-[[action]].

A [[homomorphism]] of symmetric spectra f:XYf\colon X \longrightarrow Y is

  • a sequence of maps f n:X nY nf_n \colon X_n \longrightarrow Y_n

such that

  1. each f nf_n [[intertwiner|intetwines]] the Σ(n)\Sigma(n)-[[action]];

  2. the following [[commuting diagram|diagrams commute]]

    S 1X n f nid S 1Y n σ n X σ n Y X n+1 f n+1 Y n+1. \array{ S^1 \wedge X_n &\stackrel{f_n \wedge id}{\longrightarrow}& S^1 \wedge Y_n \\ \downarrow^{\mathrlap{\sigma^X_n}} && \downarrow^{\mathrlap{\sigma^Y_n}} \\ X_{n+1} &\stackrel{f_{n+1}}{\longrightarrow}& Y_{n+1} } \,.

We write SymSpec(Top cg)SymSpec(Top_{cg}) for the resulting [[category]] of symmetric spectra.

(Hovey-Shipley-Smith 00, def. 1.2.2, Schwede 12, I, def. 1.1)

The definition of orthogonal spectra has the same structure, just with the [[symmetric groups]] replaced by the [[orthogonal groups]].

Definition

A topological [[orthogonal spectrum]] XX is

  1. a sequence {X nTop cg */|n}\{X_n \in Top_{cg}^{\ast/}\;\vert\; n \in \mathbb{N}\} of [[pointed topological space|pointed]] [[compactly generated topological spaces]];

  2. a basepoint preserving continuous right [[action]] of the [[orthogonal group]] O(n)O(n) on X nX_n;

  3. a sequence of morphisms σ n:S 1X nX n+1\sigma_n \colon S^1 \wedge X_n \longrightarrow X_{n+1}

such that

  • for all n,kn, k \in \mathbb{N} the [[composition|composite]]

    S kX nS k1S 1X nidσ nS k1X n+1S k2S 1X n+2idσ n+1σ n+k1X n+k S^{k} \wedge X_n \simeq S^{k-1} \wedge S^1 \wedge X_n \stackrel{id \wedge \sigma_n }{\longrightarrow} S^{k-1} \wedge X_{n+1} \simeq S^{k-2}\wedge S^1 \wedge X_{n+2} \stackrel{id \wedge \sigma_{n+1}}{\longrightarrow} \cdots \stackrel{\sigma_{n+k-1}}{\longrightarrow} X_{n+k}

    [[intertwiner|intertwines]] the O(n)×Ok()O(n) \times Ok()-[[action]].

A [[homomorphism]] of orthogonal spectra f:XYf\colon X \longrightarrow Y is

  • a sequence of maps f n:X nY nf_n \colon X_n \longrightarrow Y_n

such that

  1. each f nf_n [[intertwiner|intetwines]] the O(n)O(n)-[[action]];

  2. the following [[commuting diagram|diagrams commute]]

    S 1X n f nid S 1Y n σ n X σ n Y X n+1 f n+1 Y n+1. \array{ S^1 \wedge X_n &\stackrel{f_n \wedge id}{\longrightarrow}& S^1 \wedge Y_n \\ \downarrow^{\mathrlap{\sigma^X_n}} && \downarrow^{\mathrlap{\sigma^Y_n}} \\ X_{n+1} &\stackrel{f_{n+1}}{\longrightarrow}& Y_{n+1} } \,.

We write OrthSpec(Top cg)OrthSpec(Top_{cg}) for the resulting [[category]] of orthogonal spectra.

(e.g. Schwede 12, I, def. 7.2)

Proposition

Definitions and are indeed equivalent to def. :

orthogonal spectra are euqivalently the [[module objects]] over the incarnation 𝕊 orth\mathbb{S}_{orth} of the sphere spectrum

OrthSpec(Top cg)𝕊 orthMod OrthSpec(Top_{cg}) \simeq \mathbb{S}_{orth} Mod

and symmetric spectra sre equivalently the module objects over the incarnation 𝕊 sym\mathbb{S}_{sym} of the sphere spectrum

SymSpec(Top cg)𝕊 symMod. SymSpec(Top_{cg}) \simeq \mathbb{S}_{sym} Mod \,.

(Hovey-Shipley-Smith 00, prop. 2.2.1)

Proof

We discuss this for symmetric spectra. The proof for orthogonal spectra is of the same form.

First of all, by example an object in [Sym,Top cg */][Sym, Top^{\ast/}_{cg}] is equivalently a “symmetric sequence”, namely a sequence of pointed topological spaces X kX_k, for kk \in \mathbb{N}, equipped with an [[action]] of Σ(k)\Sigma(k) (def. ).

By corollary and lemma , the structure morphism of an 𝕊 sym\mathbb{S}_{sym}-[[module object]] on XX

𝕊 sym DayXX \mathbb{S}_{sym} \otimes_{Day} X \longrightarrow X

is equivalently (as a [[functor with smash products]]) a natural transformation

S n 1X n 2X n 1+n 2 S^{n_1} \wedge X_{n_2} \longrightarrow X_{n_1 + n_2}

over Sym×SymSym \times Sym. This means equivalently that there is such a morphism for all n 1,n 2n_1, n_2 \in \mathbb{N} and that it is Σ(n 1)×Σ(n 2)\Sigma(n_1) \times \Sigma(n_2)-equivariant.

Hence it only remains to see that these natural transformations are uniquely fixed once the one for n 1=1n_1 = 1 is given. To that end, observe that lemma says that in the following [[commuting squares]] (exhibiting the action property on the level of functors with smash product, where we are notationally suppressing the [[associators]]) the left vertical morphisms are [[isomorphisms]]:

S n 1S n 2X n 3 S n 1X n 2+n 3 S n 1+n 2X n 3 X n 1+n 2+n 3. \array{ S^{n_1}\wedge S^{n_2} \wedge X_{n_3} &\longrightarrow& S^{n_1} \wedge X_{n_2 + n_3} \\ {}^{\mathllap{\simeq}}\downarrow && \downarrow \\ S^{n_1+ n_2} \wedge X_{n_3} &\longrightarrow& X_{n_1 + n_2 + n_3} } \,.

This says exactly that the action of S n 1+n 2S^{n_1 + n_2} has to be the composite of the actions of S n 2S^{n_2} followed by that of S n 1S^{n_1}. Hence the statement follows by [[induction]].

Finally, the definition of [[homomorphisms]] on both sides of the equivalence are just so as to preserve precisely this structure, hence they conincide under this identification.

Definition

Given X,YSymSpec(Top cg)X,Y \in SymSpec(Top_{cg}) two [[symmetric spectra]], def. , then their [[smash product of spectra]] is the symmetric spectrum

XYSymSpec(Top cg) X \wedge Y \; \in SymSpec(Top_{cg})

with component spaces the [[coequalizer]]

p+1+q=nΣ(p+1+q) +Σ p×Σ 1×Σ qX pS 1Y qAAAArp+q=nΣ(p+q) +Σ p×Σ qX pY qcoeq(XY)(n) \underset{p+1+q = n}{\bigvee} \Sigma(p+1+q)_+ \underset{\Sigma_p \times \Sigma_1 \times \Sigma_q}{\wedge} X_p \wedge S^1 \wedge Y_q \underoverset {\underset{r}{\longrightarrow}} {\overset{\ell}{\longrightarrow}} {\phantom{AAAA}} \underset{p+q=n}{\bigvee} \Sigma(p+q)_+ \underset{\Sigma_p \times \Sigma_q}{\wedge} X_p \wedge Y_q \overset{coeq}{\longrightarrow} (X \wedge Y)(n)

where \ell has components given by the structure maps

X pS 1Y qidσ qX pY q X_p \wedge S^1 \wedge Y_q \overset{id \wedge \sigma_{q}}{\longrightarrow} X_p \wedge Y_q

while rr has components given by the structure maps conjugated by the [[braiding]] in Top cg */Top^{\ast/}_{cg} and the [[permutation]] [[action]] χ p,1\chi_{p,1} (that [[shuffle|shuffles]] the element on the right to the left)

X pS 1X qτ X p,S 1 Top cg */idS 1X pX qσ pidX p+1X qχ p,1idX 1+pX q. X_p \wedge S^1 \wedge X_q \overset{\tau^{Top^{\ast/}_{cg}}_{X_p,S^1} \wedge id}{\longrightarrow} S^1 \wedge X_p \wedge X_q \overset{\sigma_p\wedge id}{\longrightarrow} X_{p+1} \wedge X_q \overset{\chi_{p,1} \wedge id}{\longrightarrow} X_{1+p} \wedge X_q \,.

Finally The structure maps of XYX \wedge Y are those induced under the coequalizer by

S 1(X pY q)(S 1X p)Y qσ p XidX p+1Y q. S^1 \wedge (X_p \wedge Y_q \wedge) \simeq (S^1 \wedge X_p) \wedge Y_q \overset{\sigma^X_{p} \wedge id}{\longrightarrow} X_{p+1} \wedge Y_{q} \,.

Analogously for orthogonal spectra.

(Schwede 12, p. 82)

Proposition

Under the identification of prop. , the explicit [[smash product of spectra]] in def. is equivalent to the abstractly defined tensor product in def. :

in the case of [[symmetric spectra]]:

𝕊 sym \wedge \simeq \otimes_{\mathbb{S}_{sym}}

in the case of [[orthogonal spectra]]:

𝕊 orth. \wedge \simeq \otimes_{\mathbb{S}_{orth}} \,.

(Schwede 12, E.1.16)

Proof

By def. the abstractly defined tensor product of two 𝕊 sym\mathbb{S}_{sym}-modules XX and YY is the [[coequalizer]]

X Day𝕊 sym DayYAAAAρ 1(τ X,𝕊 sym Dayid)Xρ 2XYcoeqX 𝕊 symY. X \otimes_{Day} \mathbb{S}_{sym} \otimes_{Day} Y \underoverset {\underset{\rho_{1}\circ (\tau^{Day}_{X, \mathbb{S}_{sym}} \otimes id)}{\longrightarrow}} {\overset{X \otimes \rho_2}{\longrightarrow}} {\phantom{AAAA}} X \otimes Y \overset{coeq}{\longrightarrow} X \otimes_{\mathbb{S}_{sym}} Y \,.

The [[Day convolution]] product appearing here is over the category SymSym from def. . By example and unwinding the definitions, this is for any two symmetric spectra AA and BB given degreewise by the [[wedge sum]] of component spaces summing to that total degree, smashed with the symmetric group with basepoint adjoined and then quotiented by the diagonal action of the symmetric group acting on the degrees separately:

(A DayB)(n) =n 1,n 2Σ(n 1+n 2,n)={Σ(n 1+n 2,n) + ifn 1+n 2=n * otherwise +A n 1B n 1 n 1+n 2=nΣ(n 1+n 2) +O(n 1)×O(n 2)(A n 1B n 2). \begin{aligned} (A \otimes_{Day} B)(n) & = \overset{n_1,n_2}{\int} \underset{ = \left\{ \array{ \Sigma(n_1 + n_2,n)_+ & if \; n_1+n_2 = n \\ \ast & otherwise } \right. }{ \underbrace{\Sigma(n_1 + n_2, n)} }_+ \wedge A_{n_1} \wedge B_{n_1} \\ & \simeq \underset{n_1 + n_2 = n}{\bigvee} \Sigma(n_1+n_2)_+ \underset{O(n_1) \times O(n_2) }{\wedge} \left( A_{n_1} \wedge B_{n_2} \right) \end{aligned} \,.

This establishes the form of the coequalizer diagram. It remains to see that under this identification the two abstractly defined morphisms are the ones given in def. .

To see this, we apply the adjunction isomorphism between the [[Day convolution product]] and the [[external tensor product]] (cor. ) twice, to find the following sequence of equivalent incarnations of morphisms:

(X Day(𝕊 orth DayY))(n) (X DayY)(n) Z n X n 1(𝕊 sym DayY)(n 2) X n 1Y(n 2) Z n 1+n 2 (𝕊 sym DayY)(n 2) Y(n 2) Maps(X n 1,Z n 1+n 2) S n 2Y n 3 Y n 2+n 3 Maps(X n 1,Z n 1+n 2+n 3) X n 1S n 2Y n 3 X n 1Y n 2+n 3 Z n 1+n 2+n 3. \array{ \arrayopts{\rowlines{solid}} (X \otimes_{Day} ( \mathbb{S}_{orth} \otimes_{Day} Y ))(n) &\longrightarrow& (X \otimes_{Day} Y)(n) &\longrightarrow& Z_n \\ X_{n_1} \wedge (\mathbb{S}_{sym} \otimes_{Day} Y)(n'_2) &\longrightarrow& X_{n_1}\wedge Y(n'_2) &\longrightarrow& Z_{n_1 + n'_2} \\ (\mathbb{S}_{sym} \otimes_{Day} Y)(n'_2) &\longrightarrow& Y(n'_2) &\longrightarrow& Maps(X_{n_1}, Z_{n_1 + n'_2}) \\ S^{n_2} \wedge Y_{n_3} &\longrightarrow& Y_{n_2 + n_3} &\longrightarrow& Maps(X_{n_1}, Z_{n_1 + n_2 + n_3}) \\ X_{n_1} \wedge S^{n_2} \wedge Y_{n_3} &\longrightarrow& X_{n_1} \wedge Y_{n_2 + n_3} &\longrightarrow& Z_{n_1 + n_2 + n_3} } \,.

This establishes the form of the morphism \ell. By the same reasoning as in the proof of prop. , we may restrict the coequalizer to n 2=1n_2 = 1 without changing it.

The form of the morphism rr is obtained by the analogous sequence of identifications of morphisms, now with the parenthesis to the left. That it involves τ Top cg */\tau^{Top^{\ast/}_{cg}} and the permutation action τ sym\tau^{sym} as shown above follows from the formula for the braiding of the Day convolution tensor product from the proof of prop. :

τ A,B Day(n)=n 1,n 2Sym(τ n 1,n 2 Sym,n)τ A n 1,B n 2 Top cg */ \tau^{Day}_{A,B}(n) = \overset{n_1,n_2}{\int} Sym( \tau^{Sym}_{n_1,n_2}, n ) \wedge \tau^{Top^{\ast/}_{cg}}_{A_{n_1}, B_{n_2}}

by translating it to the components of the precomposition

X Day𝕊 symτ X,𝕊 sym Day𝕊 sym DayXX X \otimes_{Day} \mathbb{S}_{sym} \overset{\tau^{Day}_{X,\mathbb{S}_{sym}}}{\longrightarrow} \mathbb{S}_{sym} \otimes_{Day} X \overset{}{\longrightarrow} X

via the formula from the proof of prop. for the [[left Kan extension]] A DayBLan A¯BA \otimes_{Day} B \simeq Lan_{\otimes} A \overline{\wedge} B (prop. ):

[Sym,Top cg */](τ X,𝕊 sym Day,X) nMaps(n 1,n 2Sym(τ n 1,n 2 sym,n)τ X n 1,S n 2 Top cg */,X(n)) * n 1,n 2Maps(τ X n 1,S n 2 Top cg */,X(τ n 1,n 2 sym)) *. \begin{aligned} [Sym, Top^{\ast/}_{cg}]( \tau^{Day}_{X,\mathbb{S}_{sym}}, X) & \simeq \underset{n}{\int} Maps( \overset{n_1, n_2}{\int} Sym( \tau^{sym}_{n_1,n_2}, n ) \wedge \tau^{Top^{\ast/}_{cg}}_{X_{n_1}, S^{n_2}} , X(n) )_\ast \\ & \simeq \underset{n_1,n_2}{\int} Maps( \tau_{X_{n_1}, S^{n_2} }^{Top^{\ast/}_{cg}} , X( \tau^{sym}_{n_1,n_2} ) )_\ast \end{aligned} \,.

This last expression is the function on morphisms which precomposes components under the coend with the braiding τ X n 1,S n 2 Top cg */\tau_{X_{n_1}, S^{n_2} }^{Top^{\ast/}_{cg}} in topological spaces and postcomposes them with the image of the functor XX of the braiding in SymSym. But the braiding in SymSym is, by def. , given by the respective shuffle permutations τ n 1,n 2 sym=χ n 1,n 2\tau^{sym}_{n_1,n_2} = \chi_{n_1,n_2}, and by prop. the image of these under XX is via the given Σ n 1+n 2\Sigma_{n_1+n_2}-action on X n 1+n 2X_{n_1+n_2}.

Finally to see that the structure map is as claimed: By prop. the structure morphisms are the degree-1 components of the 𝕊 sym\mathbb{S}_{sym}-action, and by prop. the 𝕊 sym\mathbb{S}_{sym}-action on a tensor product of 𝕊 sym\mathbb{S}_{sym}-modules is induced via the action on the left tensor factor.

Definition

A commutative [[symmetric ring spectrum]] EE is

  1. a sequence of component spaces E nTop cg */E_n \in Top^{\ast/}_{cg} for nn \in \mathbb{N};

  2. a basepoint preserving continuous left [[action]] of the [[symmetric group]] Σ(n)\Sigma(n) on E nE_n;

  3. for all n 1,n 2n_1,n_2\in \mathbb{N} a multiplication map

    μ n 1,n 2:E n 1E n 2E n 1+n 2 \mu_{n_1,n_2} \;\colon\; E_{n_1} \wedge E_{n_2} \longrightarrow E_{n_1 + n_2}

    (a morphism in Top cg */Top^{\ast/}_{cg})

  4. two unit maps

    ι 0:S 0E 0 \iota_0 \;\colon\; S^0 \longrightarrow E_0
    ι 1:S 1E 1 \iota_1 \;\colon\; S^1 \longrightarrow E_1

such that

  1. (equivariance) μ n 1,n 2\mu_{n_1,n_2} [[intertwiner|intertwines]] the Σ(n 1)×Σ(n 2)\Sigma(n_1) \times \Sigma(n_2)-action;

  2. (associativity) for all n 1,n 2,n 3n_1, n_2, n_3 \in \mathbb{N} the following [[commuting diagram|diagram commutes]] (where we are notationally suppressing the [[associators]] of (Top cg */,,S 0)(Top^{\ast/}_{cg}, \wedge, S^0))

    E n 1E n 2E n 3 idμ n 2,n 3 E n 1E n 2+n 3 μ n 1,n 2id μ n 1,n 2+n 3 E n 1+n 2E n 3 μ n 1+n 2,n 3 E n 1+n 2+n 3; \array{ E_{n_1} \wedge E_{n_2} \wedge E_{n_3} &\overset{id \wedge \mu_{n_2,n_3}}{\longrightarrow}& E_{n_1} \wedge E_{n_2 + n_3} \\ {}^{\mathllap{\mu_{n_1,n_2}\wedge id }}\downarrow && \downarrow^{\mathrlap{\mu_{n_1, n_2 + n_3}}} \\ E_{n_1 + n_2} \wedge E_{n_3} &\underset{\mu_{n_1 + n_2, n_3}}{\longrightarrow}& E_{n_1 + n_2 + n_3} } \,;
  3. (unitality) for all nn \in \mathbb{N} the following [[commuting diagram|diagram commutes]]

    S 0E n ι 0id E 0E n E n Top cg */ μ 0,n E n \array{ S^0 \wedge E_n &\overset{\iota_0 \wedge id}{\longrightarrow}& E_0 \wedge E_n \\ &{}_{\mathllap{\ell^{Top^{\ast/}_{cg}}_{E_n}}}\searrow& \downarrow^{\mathrlap{\mu_{0,n}}} \\ && E_n }

    and

    E nS 0 idι 0 E nE 0 r E n Top cg */ μ n,0 E n, \array{ E_n \wedge S^0 &\overset{id \wedge \iota_0 }{\longrightarrow}& E_n \wedge E_0 \\ &{}_{\mathllap{r^{Top^{\ast/}_{cg}}_{E_n}}}\searrow& \downarrow^{\mathrlap{\mu_{n,0}}} \\ && E_n } \,,

    where the diagonal morphisms \ell and rr are the left and right [[unitors]] in (Top cg */,,S 0)(Top^{\ast/}_{cg}, \wedge, S^0), respectively.

  4. (commutativity) for all n 1,n 2n_1, n_2 \in \mathbb{N} the following [[commuting diagram|diagram commutes]]

    E n 1E n 2 τ E n 1,E n 2 Top cg */ E n 2E n 1 μ n 1,n 2 μ n 2,n 1 E n 1+n 2 χ n 1,n 2 E n 2+n 1, \array{ E_{n_1} \wedge E_{n_2} &\overset{\tau^{Top^{\ast/}_{cg}}_{E_{n_1}, E_{n_2}}}{\longrightarrow}& E_{n_2} \wedge E_{n_1} \\ {}^{\mathllap{\mu_{n_1,n_2}}}\downarrow && \downarrow^{\mathrlap{\mu_{n_2,n_1}}} \\ E_{n_1 + n_2} &\underset{\chi_{n_1,n_2}}{\longrightarrow}& E_{n_2 + n_1} } \,,

    where the top morphism τ\tau is the [[braiding]] in (Top cg */,,S 0)(Top^{\ast/}_{cg}, \wedge, S^0) (def. ) and where χ n 1,n 2Σ(n 1+n 2)\chi_{n_1,n_2} \in \Sigma(n_1 + n_2) denotes the [[permutation]] action which [[shuffle|shuffles]] the first n 1n_1 elements past the last n 2n_2 elements.

A [[homomorphism]] of symmetric commutative ring spectra f:EEf \colon E \longrightarrow E' is a sequence f n:E nE nf_n \;\colon\; E_n \longrightarrow E'_n of Σ(n)\Sigma(n)-equivariant pointed continuous functions such that the following [[commuting diagram|diagrams commute]] for all n 1,n 2n_1, n_2 \in \mathbb{N}

E n 1E n 2 f n 1f n 2 E n 1E n 2 μ n 1,n 2 μ n 2,n 1 E n 1+n 2 χ n 1,n 2 E n 2+n 1 \array{ E_{n_1} \wedge E_{n_2} &\overset{f_{n_1} \wedge f_{n_2}}{\longrightarrow}& E'_{n_1} \wedge E'_{n_2} \\ {}^{\mathllap{\mu_{n_1,n_2}}}\downarrow && \downarrow^{\mu_{n_2,n_1}} \\ E_{n_1 + n_2} &\underset{\chi_{n_1, n_2}}{\longrightarrow}& E_{n_2 + n_1} }

and f 0ι 0=ι 0f_0 \circ \iota_0 = \iota_0 and f 1ι 1=ι 1f_1\circ \iota_1 = \iota_1.

Write

CRing(SymSpec(Top cg)) CRing(SymSpec(Top_{cg}))

for the resulting [[category]] of symmetric [[commutative ring spectra]].

We regard a symmetric ring spectrum in particular as a [[symmetric spectrum]] (def. ) by taking the structure maps to be

σ n:S 1E nι 1idE 1E nμ 1,nE n+1. \sigma_n \;\colon\; S^1 \wedge E_n \overset{\iota_1 \wedge id}{\longrightarrow} E_1 \wedge E_n \overset{\mu_{1,n}}{\longrightarrow} E_{n+1} \,.

This defines a [[forgetful functor]]

CRing(SymSpec(Top cg))SymSpec(Top cg) CRing(SymSpec(Top_{cg})) \longrightarrow SymSpec(Top_{cg})

There is an analogous definition of [[orthogonal ring spectrum]] and we write

CRing(OrthSpec(Top cg)) CRing(OrthSpec(Top_{cg}))

for the category that these form.

(e.g. Schwede 12, def. 1.3)

We discuss examples below in a dedicated section Examples.

Proposition

The symmetric (orthogonal) [[commutative ring spectra]] in def. are equivalently the [[commutative monoid in a symmetric monoidal category|commutative monoids in]] (def. ) the [[symmetric monoidal category]] 𝕊 symMod\mathbb{S}_{sym}Mod (𝕊 orthMod\mathbb{S}_{orth}Mod) of def. with respect to the [[symmetric monoidal smash product of spectra]] = 𝕊 sym\wedge = \otimes_{\mathbb{S}_{sym}} (= 𝕊 orth\wedge = \otimes_{\mathbb{S}_{orth}}). Hence there are [[equivalences of categories]]

CRing(SymSpec(Top cg))CMon(𝕊 symMod, 𝕊 sym,𝕊 sym) CRing(SymSpec(Top_{cg})) \;\simeq\; CMon( \mathbb{S}_{sym}Mod, \otimes_{\mathbb{S}_{sym}}, \mathbb{S}_{sym} )

and

CRing(OrthSpec(Top cg))CMon(𝕊 orthMod, 𝕊 orth,𝕊 orth). CRing(OrthSpec(Top_{cg})) \;\simeq\; CMon( \mathbb{S}_{orth}Mod, \otimes_{\mathbb{S}_{orth}}, \mathbb{S}_{orth} ) \,.

Moreover, under these identifications the canonical forgetful functor

CMon(𝕊 symMod, 𝕊 sym,𝕊 sym)SymSpec(Top cg) CMon( \mathbb{S}_{sym}Mod, \otimes_{\mathbb{S}_{sym}}, \mathbb{S}_{sym} ) \longrightarrow SymSpec(Top_{cg})

and

CMon(𝕊 orthMod, 𝕊 orth,𝕊 orth)OrthSpec(Top cg) CMon( \mathbb{S}_{orth}Mod, \otimes_{\mathbb{S}_{orth}}, \mathbb{S}_{orth} ) \longrightarrow OrthSpec(Top_{cg})

coincides with the forgetful functor defined in def. .

Proof

We discuss this for symmetric spectra. The proof for orthogonal spectra is directly analogous.

By prop. and def. , the commutative monoids in 𝕊 symMod\mathbb{S}_{sym}Mod are equivalently commtutative monoids EE in ([Sym,Top cg */], Day,y(0))([Sym,Top^{\ast/}_{cg}], \otimes_{Day}, y(0)) equipped with a homomorphism of monoids 𝕊 symE\mathbb{S}_{sym} \longrightarrow E. In turn, by prop. this are equivalently braided lax monoidal functors (which we denote by the same symbols, for convenience) of the form

E:(Sym,+,0)(Top cg */,,S 0) E \;\colon\; (Sym, +, 0) \longrightarrow (Top^{\ast/}_{cg}, \wedge, S^0)

equipped with a [[monoidal natural transformation]] (def. )

ι:𝕊 symE. \iota \;\colon\; \mathbb{S}_{sym} \longrightarrow E \,.

The structure morphism of such a lax monoidal functor EE has as components precisely the morphisms μ n 1,n 2:E n 1E n 2E n 1+n 2\mu_{n_1, n_2}\colon E_{n_1} \wedge E_{n_2} \to E_{n_1 + n_2}. In terms of these, the associativity and braiding condition on the lax monoidal functor are manifestly the above associativity and commutativity conditions.

Moreover, by the proof of prop. the 𝕊 sym\mathbb{S}_{sym}-module structure on an an 𝕊 sym\mathbb{S}_{sym}-algebra EE has action given by

𝕊 symEeidEEμE, \mathbb{S}_{sym} \wedge E \overset{e \wedge id}{\longrightarrow} E \wedge E \overset{\mu}{\longrightarrow} E \,,

which shows, via the identification in prop , that the forgetful functors to underlying symmetric spectra coincide as claimed.

Hence it only remains to match the nature of the units. The above unit morphism ι\iota has components

ι n:S nE n \iota_n \;\colon\; S^n \longrightarrow E_n

for all nn \in \mathbb{N}, and the unitality condition for ι 0\iota_0 and ι 1\iota_1 is manifestly as in the statement above.

We claim that the other components are uniquely fixed by these:

By lemma , the product structure in 𝕊 sym\mathbb{S}_{sym} is by isomorphisms S n 1S n 2S n 1+n 2S^{n_1} \wedge S^{n_2} \simeq S^{n_1 + n_2}, so that the commuting square for the coherence condition of this [[monoidal natural transformation]]

S n 1S n 2 ι n 1ι n 2 E n 1E n 2 μ n 1,n 2 S n 1+n 2 ι n 1+n 2 E n 1+n 2 \array{ S^{n_1} \wedge S^{n_2} &\overset{\iota_{n_1} \wedge \iota_{n_2}}{\longrightarrow}& E_{n_1} \wedge E_{n_2} \\ {}^{\mathllap{\simeq}}\downarrow && \downarrow^{\mathrlap{\mu_{n_1, n_2}}} \\ S^{n_1 + n_2} &\underset{\iota_{n_1 + n_2}}{\longrightarrow}& E_{n_1 + n_2} }

says that ι n 1+n 2=μ n 1,n 2(ι n 1ι n 2)\iota_{n_1 + n_2} = \mu_{n_1,n_2} \circ (\iota_{n_1} \wedge \iota_{n_2}). This means that ι n2\iota_{n \geq 2} is uniquely fixed once ι 0\iota_0 and ι 1\iota_1 are given.

Finally it is clear that homomorphisms on both sides of the equivalence precisely respect all this structure under both sides of the equivalence.

Similarly:

Definition

Given a symmetric (orthogonal) [[commutative ring spectrum]] EE (def. ), then a left symmetric (orthogonal) [[module spectrum]] NN over EE is

  1. a sequence of component spaces N nTop cg */N_n \in Top^{\ast/}_{cg} for nn \in \mathbb{N};

  2. a basepoint preserving continuous left [[action]] of the [[symmetric group]] Σ(n)\Sigma(n) on N nN_n;

  3. for all n 1,n 2n_1,n_2\in \mathbb{N} an action map

    ρ n 1,n 2:E n 1N n 2N n 1+n 2 \rho_{n_1,n_2} \;\colon\; E_{n_1} \wedge N_{n_2} \longrightarrow N_{n_1 + n_2}

    (a morphism in Top cg */Top^{\ast/}_{cg})

such that

  1. (equivariance) ρ n 1,n 2\rho_{n_1,n_2} intertwines the Σ(n 1)×Σ(n 2)\Sigma(n_1) \times \Sigma(n_2)-action;

  2. (action property) for all n 1,n 2,n 3n_1, n_2, n_3 \in \mathbb{N} the following [[commuting diagram|diagram commutes]] (where we are notationally suppressing the [[associators]] of (Top cg */,,S 0)(Top^{\ast/}_{cg}, \wedge, S^0))

    E n 1E n 2N n 3 idρ n 2,n 3 E n 1N n 2+n 3 μ n 1,n 2id ρ n 1,n 2+n 3 E n 1+n 2N n 3 ρ n 1+n 2,n 3 N n 1+n 2+n 3; \array{ E_{n_1} \wedge E_{n_2} \wedge N_{n_3} &\overset{id \wedge \rho_{n_2,n_3}}{\longrightarrow}& E_{n_1} \wedge N_{n_2 + n_3} \\ {}^{\mathllap{\mu_{n_1,n_2}\wedge id }}\downarrow && \downarrow^{\mathrlap{\rho_{n_1, n_2 + n_3}}} \\ E_{n_1 + n_2} \wedge N_{n_3} &\underset{\rho_{n_1 + n_2, n_3}}{\longrightarrow}& N_{n_1 + n_2 + n_3} } \,;
  3. (unitality) for all nn \in \mathbb{N} the following [[commuting diagram|diagram commutes]]

    S 0N n ι 0id E 0N n N n Top cg */ μ 0,n N n. \array{ S^0 \wedge N_n &\overset{\iota_0 \wedge id}{\longrightarrow}& E_0 \wedge N_n \\ &{}_{\mathllap{\ell^{Top^{\ast/}_{cg}}_{N_n}}}\searrow& \downarrow^{\mathrlap{\mu_{0,n}}} \\ && N_n } \,.

A [[homomorphism]] of left EE-module spectra f:NNf\;\colon\; N \longrightarrow N' is a sequence of pointed continuous functions f n:N nN nf_n \;\colon\; N_n \longrightarrow N'_n such that for all n 1,n 2n_1,n_2 \in \mathbb{N} the following [[commuting diagram|diagrams commute]]:

E n 1N n 2 idf n 2 E n 1N n 2 ρ n 1,n 2 ρ n 1,n 2 N n 1+n 2 f n 1+n 2 N n 1+n 2. \array{ E_{n_1} \wedge N_{n_2} &\overset{id \wedge f_{n_2}}{\longrightarrow}& E_{n_1} \wedge N'_{n_2} \\ {}^{\mathllap{\rho_{n_1,n_2}}}\downarrow && \downarrow^{\rho_{n_1, n_2}} \\ N_{n_1 + n_2} &\underset{f_{n_1 + n_2}}{\longrightarrow}& N'_{n_1 + n_2} } \,.

We write

EMod(SymSpec(Top cg)),EMod(OrthSpec(Top cg)) E Mod(SymSpec(Top_{cg})) \;\;\,, \;\; E Mod(OrthSpec(Top_{cg}))

for the resulting category of symmetric (orthogonal) EE-module spectra.

(e.g. Schwede 12, I, def. 1.5)

Proposition

Under the identification, from prop. , of [[commutative ring spectra]] with [[commutative monoids in a symmetric monoidal category|commutative monoids with respect to]] the [[symmetric monoidal smash product of spectra]], the EE-[[module spectra]] of def. are equivalently the left [[module objects]] (def. ) over the respective monoids, i.e. there are [[equivalences of categories]]

EMod(SymSpec(Top cg))EMod([Sym,Top cg */], Day,y(0)) E Mod(SymSpec(Top_{cg})) \;\simeq\; E Mod( [Sym,Top^{\ast/}_{cg}], \otimes_{Day}, y(0) )

and

EMod(OrthSpec(Top cg))EMod([Orth,Top cg */], Day,y(0)), E Mod(OrthSpec(Top_{cg})) \;\simeq\; E Mod( [Orth, Top^{\ast/}_{cg}], \otimes_{Day}, y(0) ) \,,

where on the right we have the [[categories of modules]] from def. .

Proof

The proof is directly analogous to that of prop. . Now prop. and prop. give that the module objects in question are equivalently [[modules over a monoidal functor]] (def. ) and writing these out in components yields precisely the above structures and properties.

As diagram spectra

In [[Introduction to Stable homotopy theory – 1-1]] we obtained the strict/level [[model structure on topological sequential spectra]] by identifying the category SeqSpec(Top cg)SeqSpec(Top_{cg}) of [[sequential spectra]] with a category of [[topologically enriched functors]] with values in Top cg */Top^{\ast/}_{cg} (prop.) and then invoking the general existence of the [[projective model structure on functors]] (thm.).

Here we discuss the analogous construction for the more general structured spectra from above.

Proposition

Let (𝒞, 𝒞,1 𝒞)(\mathcal{C},\otimes_{\mathcal{C}}, 1_{\mathcal{C}}) be a [[topologically enriched category|topologically enriched]] [[monoidal category]] (def. ), and let AMon([𝒞,Top cg */], Day,y(1 𝒞))A \in Mon([\mathcal{C},Top^{\ast/}_{cg}],\otimes_{Day}, y(1_{\mathcal{C}})) be a [[monoid in a monoidal category|monoid in]] (def. ) the pointed topological [[Day convolution]] monoidal category over 𝒞\mathcal{C} from prop. .

Then the [[category of modules|category of left A-modules]] (def. ) is a pointed topologically [[enriched functor category]] category (exmpl.)

AMod[AFree 𝒞Mod op,Top cg */], A Mod \;\simeq\; [ A Free_{\mathcal{C}}Mod^{op}, \; Top_{cg}^{\ast/} ] \,,

over the category of [[free modules]] over AA (prop. ) on objects in 𝒞\mathcal{C} (under the [[Yoneda embedding]] y:𝒞 op[𝒞,Top cg */]y \colon \mathcal{C}^{op} \to [\mathcal{C}, Top^{\ast/}_{cg}]). Hence the objects of AFree 𝒞ModA Free_{\mathcal{C}}Mod are identified with those of 𝒞\mathcal{C}, and its [[hom-spaces]] are

AFree 𝒞Mod(c 1,c 2)=AMod(A Dayy(c 1),A Dayy(c 2)). A Free_{\mathcal{C}}Mod( c_1, c_2) \;=\; A Mod( A \otimes_{Day} y(c_1),\; A \otimes_{Day} y(c_2) ) \,.

(MMSS 00, theorem 2.2)

Proof

Use the identification from prop. of AA with a [[lax monoidal functor]] and of any AA-[[module object]] NN as a functor with the structure of a [[module over a monoidal functor]], given by [[natural transformations]]

A(c 1)N(c 2)ρ c 1,c 2N(c 1c 2). A(c_1)\otimes N(c_2) \overset{\rho_{c_1,c_2}}{\longrightarrow} N(c_1 \otimes c_2) \,.

Notice that these transformations have just the same structure as those of the [[enriched functor|enriched functoriality]] of NN (def.) of the form

𝒞(c 1,c 2)N(c 1)N(c 2). \mathcal{C}(c_1,c_2) \otimes N(c_1) \overset{}{\longrightarrow} N(c_2) \,.

Hence we may unify these two kinds of transformations into a single kind of the form

𝒞(c 3c 1,c 2)A(c 3)N(c 1)idρ c 3,c 1𝒞(c 3c 1,c 2)N(c 3c 2)N(c 2) \mathcal{C}(c_3 \otimes c_1 , c_2) \otimes A(c_3) \otimes N(c_1) \overset{id \otimes \rho_{c_3,c_1}}{\longrightarrow} \mathcal{C}(c_3 \otimes c_1, c_2) \otimes N(c_3 \otimes c_2) \longrightarrow N(c_2)

and subject to certain identifications.

Now observe that the hom-objects of AFree 𝒞ModA Free_{\mathcal{C}}Mod have just this structure:

AFree 𝒞Mod(c 2,c 1) =AMod(A Dayy(c 2),A Dayy(c 1)) [𝒞,Top cg */](y(c 2),A Dayy(c 1)) (A Dayy(c 1))(c 2) c 3,c 4𝒞(c 3c 4,c 2)A(c 3)𝒞(c 1,c 4) c 3𝒞(c 3c 1,c 2)A(c 3). \begin{aligned} A Free_{\mathcal{C}}Mod(c_2,c_1) & = A Mod( A \otimes_{Day} y(c_2) , A \otimes_{Day} y(c_1) ) \\ & \simeq [\mathcal{C},Top^{\ast/}_{cg}](y(c_2), A \otimes_{Day} y(c_1) ) \\ & \simeq (A \otimes_{Day} y(c_1) )(c_2) \\ & \simeq \overset{c_3,c_4}{\int} \mathcal{C}(c_3 \otimes c_4,c_2) \wedge A(c_3) \wedge \mathcal{C}(c_1, c_4) \\ & \simeq \overset{c_3}{\int} \mathcal{C}(c_3 \otimes c_1, c_2) \wedge A (c_3) \end{aligned} \,.

Here we used first the [[free-forgetful adjunction]] of prop. , then the [[enriched Yoneda lemma]] (prop. ), then the [[coend]]-expression for [[Day convolution]] (def. ) and finally the [[co-Yoneda lemma]] (prop. ).

Then define a [[topologically enriched category]] 𝒟\mathcal{D} to have [[objects]] and [[hom-spaces]] those of AFree 𝒞Mod opA Free_{\mathcal{C}}Mod^{op} as above, and whose [[composition]] operation is defined as follows:

𝒟(c 2,c3)𝒟(c 1,c 2) (c 5𝒞(c 5 𝒞c 2,c 3)A(c 5))(c 4𝒞(c 4 𝒞c 1,c 2)A(c 4)) c 4,c 5𝒞(c 5 𝒞c 2,c 3)𝒞(c 4 𝒞c 1,c 2)A(c 5)A(c 4) c 4,c 5𝒞(c 5 𝒞c 2,c 3)𝒞(c 5 𝒞c 4 𝒞c 1,c 5 𝒞c 2)A(c 5 𝒞c 4) c 4,c 5𝒞(c 5 𝒞c 4 𝒞c 1,c 5 𝒞c 2)A(c 5 𝒞c 4) c 4𝒞(c 4 𝒞c 1,c 3) VA(c 4), \begin{aligned} \mathcal{D}(c_2,c3) \wedge \mathcal{D}(c_1,c_2) & \simeq \left( \overset{c_5}{\int} \mathcal{C}(c_5 \otimes_{\mathcal{C}} c_2 , c_3 ) \wedge A(c_5) \right) \wedge \left( \overset{c_4}{\int} \mathcal{C}(c_4 \otimes_{\mathcal{C}} c_1, c_2) \wedge A(c_4) \right) \\ & \simeq \overset{c_4, c_5}{\int} \mathcal{C}(c_5 \otimes_{\mathcal{C}} c_2 , c_3) \wedge \mathcal{C}(c_4 \otimes_{\mathcal{C}} c_1, c_2 ) \wedge A(c_5) \wedge A(c_4) \\ & \longrightarrow \overset{c_4,c_5}{\int} \mathcal{C}(c_5 \otimes_{\mathcal{C}} c_2 , c_3) \wedge \mathcal{C}(c_5 \otimes_{\mathcal{C}} c_4 \otimes_{\mathcal{C}} c_1, c_5 \otimes_{\mathcal{C}} c_2 ) \wedge A(c_5 \otimes_{\mathcal{C}} c_4 ) \\ & \longrightarrow \overset{c_4, c_5}{\int} \mathcal{C}(c_5 \otimes_{\mathcal{C}} c_4 \otimes_{\mathcal{C}} c_1, c_5 \otimes_{\mathcal{C}} c_2 ) \wedge A(c_5 \otimes_{\mathcal{C}} c_4 ) \\ & \longrightarrow \overset{c_4}{\int} \mathcal{C}(c_4 \otimes_{\mathcal{C}} c_1 , c_3) \otimes_V A(c_4 ) \end{aligned} \,,

where

  1. the equivalence is [[braiding]] in the integrand (and the [[Fubini theorem]], prop. );

  2. the first morphism is, in the integrand, the smash product of

    1. forming the tensor product of hom-objects of 𝒞\mathcal{C} with the identity morphism on c 5c_5;

    2. the monoidal functor incarnation A(c 5)A(c 4)A(c 5 𝒞c 4)A(c_5) \wedge A(c_4)\longrightarrow A(c_5 \otimes_{\mathcal{C}} c_4 ) of the monoid structure on AA;

  3. the second morphism is, in the integrand, given by composition in 𝒞\mathcal{C};

  4. the last morphism is the morphism induced on [[coends]] by regarding [[extranatural transformation|extranaturality]] in c 4c_4 and c 5c_5 separately as a special case of extranaturality in c 6c 4c 5c_6 \coloneqq c_4 \otimes c_5 (and then renaming).

With this it is fairly straightforward to see that

AMod[𝒟,Top cg */], A Mod \simeq [\mathcal{D}, Top^{\ast/}_{cg}] \,,

because, by the above definition of composition, functoriality over 𝒟\mathcal{D} manifestly encodes the AA-[[action]] property together with the functoriality over 𝒞\mathcal{C}.

This way we are reduced to showing that actually 𝒟AFree 𝒞Mod op\mathcal{D} \simeq A Free_{\mathcal{C}}Mod^{op}.

But by construction, the image of the objects of 𝒟\mathcal{D} under the [[Yoneda embedding]] are precisely the free AA-modules over objects of 𝒞\mathcal{C}:

𝒟(c,)AFree 𝒞Mod(,c)(A Dayy(c))(). \mathcal{D}(c,-) \simeq A Free_{\mathcal{C}}Mod(-,c) \simeq (A \otimes_{Day} y(c))(-) \,.

Since the [[Yoneda embedding]] is [[fully faithful functor|fully faithful]], this shows that indeed

𝒟 opAFree 𝒞ModAMod. \mathcal{D}^{op} \simeq A Free_{\mathcal{C}}Mod \hookrightarrow A Mod \,.
Example

For the sequential case Dia=SeqDia = Seq in def. , then the opposite category of [[free modules]] on objects in SeqSeq over 𝕊 seq\mathbb{S}_{seq} (def.) is identified as the category StdSpheresStdSpheres (def.):

𝕊 seqFree seqMod opStdSpheres \mathbb{S}_{seq} Free_{seq}Mod^{op} \;\simeq\; StdSpheres

Accordingly, in this case prop. reduces to the identification (prop.) of [[sequential spectra]] as topological diagrams over StdSpheresStdSpheres:

[𝕊 seqFree seqMod op,Top cg */][StdSpheres,Top cg */]SeqSpec(Top cg). [ \mathbb{S}_{seq} Free_{seq}Mod^{op}, Top^{\ast/}_{cg} ] \simeq [StdSpheres, Top^{\ast/}_{cg}] \simeq SeqSpec(Top_{cg}) \,.
Proof

There is one object y(n)y(n) for each nn \in \mathbb{N}. Moreover, from the expression in the proof of prop. we compute the [[hom-spaces]] between these to be

𝕊 seqFree seqMod(𝕊 seq Dayy k 2,𝕊 seq Dayy k 1) nSeq(n+k 1,k 2)𝕊 seq(n) {S k 2k 1 ifk 2k 1 * otherwise. \begin{aligned} \mathbb{S}_{seq} Free_{seq}Mod( \mathbb{S}_{seq} \otimes_{Day} y_{k_2} , \mathbb{S}_{seq} \otimes_{Day} y_{k_1} ) & \simeq \overset{n}{\int} Seq(n + k_1 , k_2) \wedge \mathbb{S}_{seq}(n) \\ & \simeq \left\{ \array{ S^{k_2-k_1} & if \; k_2 \geq k_1 \\ \ast & otherwise } \right. \end{aligned} \,.

These are the objects and hom-spaces of the category StdSpheresStdSpheres. It is straightforward to check that the definition of composition agrees, too.

Stable weak homotopy equivalences

We consider the evident version of [[stable weak homotopy equivalences]] for [[structured spectra]] and prove a few technical lemmas about them that are needed in the proof of the stable model structure below

Definition

For Dia{Top cg,fin */,Orth,Sym,Seq}Dia \in \{Top^{\ast/}_{cg,fin}, Orth, Sym, Seq\} one of the shapes of structured spectra from def. , let 𝕊 diaMod\mathbb{S}_{dia}Mod be the corresponding category of structured spectra (def. , prop. , def. ).

  1. The [[stable homotopy groups]] of an object X𝕊 diaModX \in \mathbb{S}_{dia}Mod are those of the underlying sequential spectrum (def.):

    π (X)π (seq *X). \pi_\bullet(X) \coloneqq \pi_\bullet(seq^\ast X) \,.
  2. An object X𝕊 diaModX \in \mathbb{S}_{dia}Mod is a structured [[Omega-spectrum]] if the underlying [[sequential spectrum]] seq *Xseq^\ast X (def. ) is a sequential [[Omega spectrum]] (def.)

  3. A morphism ff in 𝕊 diaMod\mathbb{S}_{dia}Mod is a [[stable weak homotopy equivalence]] (or: π \pi_\bullet-isomorphism) if the underlying morphism of [[sequential spectra]] seq *(f)seq^\ast(f) is a [[stable weak homotopy equivalence]] of sequential spectra (def.);

  4. a morphism ff is a stable cofibration if it is a cofibration in the strict model structure OrthSpec(Top cg) strictOrthSpec(Top_{cg})_{strict} from prop. .

(MMSS 00, def. 8.3 with the notation from p. 21, Mandell-May 02, III, def. 3.1, def. 3.2)

Lemma

Given a morphism f:XYf\;\colon\; X \longrightarrow Y in 𝕊 diaMod\mathbb{S}_{dia}Mod, then there are [[long exact sequences]] of [[stable homotopy groups]] (def. ) of the form

π +1(Y)π (Path *(f))π (X)f *π (Y)π 1(Path *(f)) \cdots \longrightarrow \pi_{\bullet + 1}(Y) \overset{}{\longrightarrow} \pi_\bullet(Path_\ast(f)) \overset{}{\longrightarrow} \pi_\bullet(X) \overset{f_\ast}{\longrightarrow} \pi_\bullet(Y) \longrightarrow \pi_{\bullet-1}(Path_\ast(f)) \longrightarrow \cdots

and

π +1(Y)π +1(Cone(f))π (X)f *π (Y)π (Cone(f)), \cdots \longrightarrow \pi_{\bullet+1}(Y) \overset{}{\longrightarrow} \pi_{\bullet+1}(Cone(f)) \overset{}{\longrightarrow} \pi_\bullet(X) \overset{f_\ast}{\longrightarrow} \pi_\bullet(Y) \longrightarrow \pi_{\bullet}(Cone(f)) \longrightarrow \cdots \,,

where Cone(f)Cone(f) denotes the [[mapping cone]] and Path *(f)Path_\ast(f) the [[mapping cocone]] of ff (def.) formed with respect to the standard [[cylinder spectrum]] X(I +)X \wedge (I_+) hence formed degreewise with respect to the standard [[reduced cylinder]] of pointed topological spaces.

(MMSS 00, theorem 7.4 (vi))

Proof

Since limits and colimits in the diagram category 𝕊 diaMod\mathbb{S}_{dia}Mod are computed objectwise, the functor seq *seq^\ast that restricts 𝕊 dia\mathbb{S}_{dia}-modules to their underlying [[sequential spectra]] preserves both limits and colimits, hence it is sufficient to consider the statement for sequential spectra.

For the first case, there is degreewise the [[long exact sequence of homotopy groups]] to the left of pointed topological spaces (exmpl.)

π 2(Y)π 1(Path *(f))π 1(X)f *π 1(Y)π 0(Path *(f))π 0(X n)f *π 0(Y n). \cdots \to \pi_2(Y) \longrightarrow \pi_1(Path_\ast(f)) \longrightarrow \pi_1(X) \overset{f_\ast}{\longrightarrow} \pi_1(Y) \longrightarrow \pi_0(Path_\ast(f)) \longrightarrow \pi_0(X_n) \overset{f_\ast}{\longrightarrow} \pi_0(Y_n) \,.

Observe that the [[sequential colimit]] that defines the [[stable homotopy groups]] (def.) preserves [[exact sequences]] of [[abelian groups]], because generally [[filtered colimits]] in [[Ab]] are [[exact functors]] (prop.). This implies that by taking the colimit over nn in the above sequences, we obtain a long exact sequence of stable homotopy groups as shown.

Now use that in sequential spectra the canonical morphism morphism Path *(f)ΩCone(f)Path_\ast(f) \longrightarrow \Omega Cone(f) is a stable weak homotopy equivalence and is compatible with the map ff (prop.) so that there is a commuting diagram of the form

π +1(Y) π (Path *(f)) π (X) f * π (Y) π 1(Path *(f)) = = = π +1(Y) π +1(Cone(f)) π (X) f * π (Y) π (Cone(f)) . \array{ \cdots &\longrightarrow& \pi_{\bullet + 1}(Y) &\overset{}{\longrightarrow}& \pi_\bullet(Path_\ast(f)) &\overset{}{\longrightarrow}& \pi_\bullet(X) &\overset{f_\ast}{\longrightarrow}& \pi_\bullet(Y) &\longrightarrow& \pi_{\bullet-1}(Path_\ast(f)) &\longrightarrow& \cdots \\ && \downarrow^{\mathrlap{=}} && \downarrow^{\mathrlap{\simeq}} && \downarrow^{\mathrlap{=}} && \downarrow^{\mathrlap{=}} && \downarrow^{\mathrlap{\simeq}} \\ \cdots &\longrightarrow& \pi_{\bullet + 1}(Y) &\overset{}{\longrightarrow}& \pi_{\bullet+1}(Cone(f)) &\overset{}{\longrightarrow}& \pi_\bullet(X) &\overset{f_\ast}{\longrightarrow}& \pi_\bullet(Y) &\longrightarrow& \pi_{\bullet}(Cone(f)) &\longrightarrow& \cdots } \,.

Since the top sequence is exact, and since all vertical morphisms are isomorphisms, it follows that also the bottom sequence is exact.

Lemma

For KTop cg,fin */K \in Top^{\ast/}_{cg,fin} a [[CW-complex]] then the operation of smash tensoring ()K(-) \wedge K preserves [[stable weak homotopy equivalences]] in 𝕊 diaMod\mathbb{S}_{dia}Mod.

Proof

Since limits and colimits in the diagram category 𝕊 diaMod\mathbb{S}_{dia}Mod are computed objectwise, the functor seq *seq^\ast that restricts 𝕊 dia\mathbb{S}_{dia}-modules to their underlying [[sequential spectra]] preserves both limits and colimits, and it also preserves smash tensoring. Hence it is sufficient to consider the statement for sequential spectra.

Fist consider the case of a finite cell complex KK.

Write

*=K 0K iK i+1K \ast = K_0 \hookrightarrow \cdots \hookrightarrow K_i \hookrightarrow K_{i+1} \hookrightarrow \cdots \hookrightarrow K

for the stages of the [[cell complex]] KK, so that for each ii there is a [[pushout]] diagram in Top cg Top^{}_{cg} of the form

S n i1 K i * (po) (po) D n i1 K i+1 S n i. \array{ S^{n_i-1} &\longrightarrow& K_i &\longrightarrow& \ast \\ {}^{\mathllap{}}\downarrow &(po)& \downarrow &(po)& \downarrow \\ D^{n_i-1} &\longrightarrow& K_{i+1} &\longrightarrow& S^{n_i} } \,.

Equivalently these are pushoutdiagrams in Top cg */Top^{\ast/}_{cg} of the form

S + n i1 K i * (po) (po) D + n i1 K i+1 S n i. \array{ S^{n_i-1}_+ &\longrightarrow& K_i &\longrightarrow& \ast \\ {}^{\mathllap{}}\downarrow &(po)& \downarrow &(po)& \downarrow \\ D^{n_i-1}_+ &\longrightarrow& K_{i+1} &\longrightarrow& S^{n_i} } \,.

Notice that it is indeed S n iS^{n_i} that appears in the top right, not S + n iS^{n_i}_+.

Now forming the smash [[tensoring]] of any morphism f:XYf\colon X \longrightarrow Y in 𝕊 diaMod(Top cg)\mathbb{S}_{dia}Mod(Top_{cg}) by the morphisms in the pushout on the right yields a commuting diagram in 𝕊 diaMod\mathbb{S}_{dia}Mod of the form

XK i XK i+1 XS n i YK i YK i+1 YS n i. \array{ X \wedge K_i &\longrightarrow& X \wedge K_{i+1} &\longrightarrow& X \wedge S^{n_i} \\ \downarrow && \downarrow && \downarrow \\ Y \wedge K_i &\longrightarrow& Y \wedge K_{i+1} &\longrightarrow& Y \wedge S^{n_i} } \,.

Here the horizontal morphisms on the left are degreewise cofibrations in Top cg */Top^{\ast/}_{cg}, hence the morphism on the right is degreewise their homotopy cofiber. This way lemma implies that there are commuting diagrams

π +1(XS n i) π (XK i) π (XK i+1) π (XS n i) π 1(XK i) fK i+1 π +1(YS n i) π (YK i) π (YK i+1) π (YS n i) π 1(XK i), \array{ \pi_{\bullet+1}(X \wedge S^{n_i}) &\longrightarrow& \pi_\bullet(X \wedge K_i) &\longrightarrow& \pi_\bullet(X \wedge K_{i+1}) &\longrightarrow& \pi_\bullet(X \wedge S^{n_i}) &\longrightarrow& \pi_{\bullet-1}(X \wedge K_i) \\ \downarrow && \downarrow && \downarrow^{\mathrlap{f \wedge K_{i+1}}} && \downarrow && \downarrow \\ \pi_{\bullet+1}(Y \wedge S^{n_i}) &\longrightarrow& \pi_\bullet(Y \wedge K_i) &\longrightarrow& \pi_\bullet(Y \wedge K_{i+1}) &\longrightarrow& \pi_\bullet(Y \wedge S^{n_i}) &\longrightarrow& \pi_{\bullet-1}(X \wedge K_i) } \,,

where the top and bottom are [[long exact sequences]] of [[stable homotopy groups]].

Now proceed by [[induction]]. For i=0i = 0 then clearly smash tensoring with K 0=*K_0 = \ast preserves stable weak homotopy equivalences. So assume that smash tensoring with K iK_i does, too. Observe that ()S n(-)\wedge S^n preserves stable weak homotopy equivalences, since ΣX[1]X\Sigma X[1]\to X is a stable weak homotopy equivalence (lemma). Hence in the above the two vertical morphisms on the left and the two on the right are isomorphism. Now the [[five lemma]] implies that also fK i+1f \wedge K_{i+1} is an isomorphism.

Finally, the statement for a non-finite cell complex follows with these arguments and then using that spheres are [[compact topological spaces|compact]] and hence maps out of them into a [[transfinite composition]] factor through some finite stage (prop.).

Lemma

The pushout in 𝕊 diaMod\mathbb{S}_{dia}Mod of a [[stable weak homotopy equivalence]] along a morphism that is degreewise a cofibration in (Top cg */) Quillen(Top^{\ast/}_{cg})_{Quillen} is again a stable weak homotopy equivalence.

Proof

Given a pushout square

X g Z f (po) Y YXZ \array{ X &\overset{g}{\longrightarrow}& Z \\ {}^{\mathllap{f}}\downarrow &(po)& \downarrow \\ Y &\underset{}{\longrightarrow}& Y \underset{X}{\sqcup}Z }

observe that the [[pasting law]] implies an isomorphism between the horizontal [[cofibers]]

X g Z cofib(g) f (po) Y YXZ cofib(g). \array{ X &\overset{g}{\longrightarrow}& Z &\longrightarrow& cofib(g) \\ {}^{\mathllap{f}}\downarrow &(po)& \downarrow && \downarrow^{\mathrlap{\simeq}} \\ Y &\underset{}{\longrightarrow}& Y \underset{X}{\sqcup}Z &\longrightarrow& cofib(g) } \,.

Moreover, since cofibrations in (Top cg */) Quillen(Top^{\ast/}_{cg})_{Quillen} are preserves by pushout, and since pushout of spectra are computed degreewise, both the top and the bottom horizontal sequences here are degreewise homotopy cofiber sequence in (Top cg */) Quillen(Top^{\ast/}_{cg})_{Quillen}. Hence lemma applies and gives a commuting diagram

π +1(cofib(g)) π (X) π (Z) π (cofib(g)) π 1(X) π (f) π +1(cofib(g)) π (Y) π (YXZ) π (cofib(g)) π 1(Y), \array{ \pi_{\bullet+1}(cofib(g)) &\longrightarrow& \pi_\bullet(X) &\overset{}{\longrightarrow}& \pi_\bullet(Z) &\longrightarrow& \pi_\bullet(cofib(g)) &\longrightarrow& \pi_{\bullet-1}(X) \\ \downarrow^{\mathrlap{\simeq}} && {}^{\mathllap{\pi_\bullet(f)}}_{\mathllap{\simeq}}\downarrow && \downarrow && \downarrow^{\mathrlap{\simeq}} && \downarrow^{\mathrlap{\simeq}} \\ \pi_{\bullet+1}(cofib(g)) &\longrightarrow& \pi_\bullet(Y) &\underset{}{\longrightarrow}& \pi_\bullet(Y \underset{X}{\sqcup}Z) &\longrightarrow& \pi_\bullet(cofib(g)) &\longrightarrow& \pi_{\bullet-1}(Y) } \,,

where the top and the bottom row are both [[long exact sequences]] of [[stable homotopy groups]]. Hence the claim now follows by the [[five lemma]].

Free spectra and Suspension spectra

The concept of [[free spectrum]] is a generalization of that of [[suspension spectrum]]. In fact the [[stable homotopy types]] of free spectra are precisely those of iterated [[loop space objects]] of [[suspension spectra]]. But for the development of the theory what matters is free spectra before passing to stable homotopy types, for as such they play the role of the basic cells for the stable [[model structures on spectra]] analogous to the role of the [[n-spheres]] in the [[classical model structure on topological spaces]] (def. below).

Moreover, while free [[sequential spectra]] are just re-indexed [[suspension spectra]], free [[symmetric spectra]] and free [[orthogonal spectra]] in addition come with suitably freely generated [[actions]] of the [[symmetric group]] and the [[orthogonal group]]. It turns out that this is not entirely trivial; it leads to a subtle issue (lemma below) where the [[adjuncts]] of certain canonical inclusions of free spectra are [[stable weak homotopy equivalences]] for sequential and orthogonal spectra, but not for symmetric spectra.

Definition

For Dia{Top fin */,Orth,Sym,Seq}Dia \in \{Top^{\ast/}_{fin}, Orth, Sym, Seq\} any one of the four diagram shapes of def. , and for each nn \in \mathbb{N}, the functor

() n:𝕊 diaModseq *𝕊 seqModSeqSpec(Top cg)() nTop cg */ (-)_n \;\colon\; \mathbb{S}_{dia}Mod \overset{seq^\ast}{\longrightarrow} \mathbb{S}_{seq}Mod \simeq SeqSpec(Top_{cg}) \stackrel{(-)_n}{\longrightarrow} Top^{\ast/}_{cg}

that sends a [[structured spectrum]] to the nnth component space of its underlying [[sequential spectrum]] has, by prop. , a [[left adjoint]]

F n dia:Top */𝕊 diaMod. F^{dia}_n \;\colon\; Top^{\ast/} \longrightarrow \mathbb{S}_{dia}Mod \,.

This is called the [[free structured spectrum]]-functor.

For the special case n=0n = 0 it is also called the structured [[suspension spectrum]] functor and denoted

Σ dia KF 0 diaK \Sigma^\infty_{dia} K \;\coloneqq\; F^{dia}_0 K

(Hovey-Shipley-Smith 00, def. 2.2.5, MMSS 00, section 8)

Lemma

Let Dia{Top fin */,Orth,Sym,Seq}Dia \in \{Top^{\ast/}_{fin}, Orth, Sym, Seq\} be any one of the four diagram shapes of def. . Then

  1. the [[free spectrum]] on KTop cg */K \in Top^{\ast/}_{cg} (def. ) is equivalently the smash [[tensoring]] with KK (def.) of the [[free module]] (def. ) over 𝕊 dia\mathbb{S}_{dia} (remark ) on the [[representable functor|representable]] y(n)[Dia,Top cg */]y(n) \in [Dia, Top^{\ast/}_{cg}]

    F n diaK (𝕊 dia Dayy(n))K 𝕊 dia Day(y(n)K); \begin{aligned} F^{dia}_n K & \simeq (\mathbb{S}_{dia} \otimes_{Day} y(n)) \wedge K \\ & \simeq \mathbb{S}_{dia} \otimes_{Day} (y(n) \wedge K) \end{aligned} \,;
  2. on nDia opy[Dia,Top cg */]n' \in Dia^{op} \stackrel{y}{\hookrightarrow} [Dia, Top^{\ast/}_{cg}] its value is given by the following [[coend]] expression (def. )

(F n diaK)(n)n 1DiaDia(n 1n,n)S n 1K. (F^{dia}_n K)(n') \;\simeq\; \overset{n_1 \in Dia}{\int} Dia(n_1 \otimes n, n') \wedge S^{n_1} \wedge K \,.

In particular the structured [[sphere spectrum]] is the free spectrum in degree 0 on the [[0-sphere]]:

𝕊 diaF 0 diaS 0 \mathbb{S}_{dia} \simeq F_0^{dia} S^0

and generally for KTop cg */K \in Top^{\ast/}_{cg} then

F 0 diaK𝕊 diaK F_0^{dia} K \simeq \mathbb{S}_{dia} \wedge K

is the smash tensoring of the strutured sphere spectrum with KK.

(Hovey-Shipley-Smith 00, below def. 2.2.5, MMSS00, p. 7 with theorem 2.2)

Proof

Under the [[equivalence of categories]]

𝕊 diaMod[𝕊 diaFree diaMod op,Top cg */] \mathbb{S}_{dia} Mod \simeq [\mathbb{S}_{dia}Free_{dia}Mod^{op}, Top^{\ast/}_{cg}]

from prop. , the expression for F n diaKF^{dia}_n K is equivalently the smash tensoring with KK of the functor that nn represents over 𝕊 diaFree diaMod\mathbb{S}_{dia}Free_{dia}Mod:

F n diaK y 𝕊 diaFree DiaMod(n)K 𝕊 diaFree diaMod(,𝕊 diay Dia(n))K \begin{aligned} F^{dia}_n K & \simeq y_{\mathbb{S}_{dia} Free_{Dia}Mod}(n) \wedge K \\ & \simeq \mathbb{S}_{dia}Free_{dia}Mod( - , \mathbb{S}_{dia} \wedge y_{Dia}(n) ) \wedge K \end{aligned}

(by [[full subcategory|fully faithfulness]] of the [[Yoneda embedding]]).

This way the first statement is a special case of the following general fact: For 𝒞\mathcal{C} a pointed [[topologically enriched category]], and for c𝒞c \in \mathcal{C} any [[object]], then there is an [[adjunction]]

[𝒞,Top cg */]() cy(c)()Top cg */ [\mathcal{C}, Top^{\ast/}_{cg}] \underoverset {\underset{(-)_c}{\longrightarrow}} {\overset{y(c)\wedge(-)}{\longleftarrow}} {\bot} Top^{\ast/}_{cg}

(saying that evaluation at cc is [[right adjoint]] to smash tensoring the functor represented by cc) witnessed by the following composite [[natural isomorphism]]:

[𝒞,Top cg */](y(c)K,F)Maps(K,[𝒞,Top cg */](y(c),F)) *Maps(K,F(c)) *=Top cg */(K,F(c)). [\mathcal{C}, Top^{\ast/}_{cg}](y(c)\wedge K, F) \;\simeq\; Maps(K, [\mathcal{C}, Top^{\ast/}_{cg}](y(c), F) )_\ast \;\simeq\; Maps(K,F(c))_\ast \;=\; Top^{\ast/}_{cg}(K,F(c)) \,.

The first is the characteristic isomorphism of [[tensoring]] from prop. , while the second is the [[enriched Yoneda lemma]] of prop. .

From this, the second statement follows by the proof of prop. .

For the last statement it is sufficient to observe that y(0)y(0) is the [[tensor unit]] under [[Day convolution]] by prop. (since 00 is the tensor unit in DiaDia), so that

F 0 diaS 0 =𝕊 dia Day(y(0)S 0) 𝕊 diay(S 0) 𝕊 dia. \begin{aligned} F_0^{dia} S^0 & = \mathbb{S}_{dia} \otimes_{Day} (y(0) \wedge S^0) \\ & \simeq \mathbb{S}_{dia} \otimes y(S^0) \\ & \simeq \mathbb{S}_{dia} \end{aligned} \,.
Proposition

Explicitly, the [[free spectra]] according to def. , look as follows:

For [[sequential spectra]]:

(F n SeqK) q{S qnK ifqn * otherwise} (F^{Seq}_n K)_q \simeq \left\{ \array{ S^{q-n} \wedge K & if \; q \geq n \\ \ast & \otherwise } \right\}

for [[symmetric spectra]]:

(F n SymK) q{Σ(q) + Σ(qn)S qnK ifqn * otherwise (F^{Sym}_n K)_q \simeq \left\{ \array{ \Sigma(q)_+ \wedge_{\Sigma(q-n)} S^{q-n} \wedge K & if\; q \geq n \\ \ast & otherwise } \right.

for [[orthogonal spectra]]:

(F n OrthK) q{O(q) + O(qn)S qnK ifqn * otherwise, (F^{Orth}_n K)_q \simeq \left\{ \array{ O(q)_+ \wedge_{O(q-n)} \wedge S^{q-n} \wedge K & if \; q \geq n \\ \ast & otherwise } \right. \,,

where “ G\wedge_G” is as in example .

(e.g. Schwede 12, example 3.20)

Proof

With the formula in item 2 of lemma we have for the case of [[orthogonal spectra]]

(F n OrthK)( q) n 1OrthOrth(n 1+n,q)={O(q) + ifn 1+n=q * otherwiseS n 1K {n 1=*B(O(qn))O(q) +O(qn)S qnK ifqn * otherwise, \begin{aligned} (F_n^{Orth} K)(\mathbb{R}^q) & \simeq \overset{n_1 \in Orth}{\int} \underset{= \left\{ \array{ O(q)_+ & if \, n_1+n = q \\ \ast & otherwise} \right.}{\underbrace{Orth(n_1 + n,q)}} \wedge S^{n_1} \wedge K \\ & \simeq \left\{ \array{ \overset{n_1 = \ast \in \mathbf{B}(O(q-n))}{\int} O(q)_+ \underset{O(q-n)}{\wedge} S^{q-n} \wedge K & if \; q \geq n \\ \ast & otherwise } \right. \end{aligned} \,,

where in the second line we used that the [[coend]] collapses to n 1=qnn_1 = q-n ranging in the full subcategory

B(O(qn) +)Orth \mathbf{B}(O(q-n)_+) \hookrightarrow Orth

on the object qn\mathbb{R}^{q-n} and then we applied example . The case of symmetric spectra is verbatim the same, with the symmetric group replacing the orthogonal group, and the case of sequential spectra is again verbatim the same, with the orthogonal group replaced by the trivial group.

Lemma

For Dia{Orth,Sym,Seq}Dia \in \{ Orth, Sym, Seq\} the diagram shape for [[orthogonal spectra]], [[symmetric spectra]] or [[sequential spectra]], then the [[free structured spectra]]

F n diaS 0𝕊 diaMod F^{dia}_n S^0 \in \mathbb{S}_{dia}Mod

from def. have component spaces that admit the structure of [[CW-complexes]].

Proof

We consider the case of [[orthogonal spectra]]. The case of [[symmetric spectra]] works verbatim the same, and the case of [[sequential spectra]] is tivial.

By prop. we have to show that for all qnq \geq n \in \mathbb{N} the topological spaces of the form

O(q) + O(qn)S qnTop cg */ O(q)_+ \wedge_{O(q-n)} S^{q-n} \;\; \in Top^{\ast/}_{cg}

admit the structure of CW-complexes.

To that end, use the [[homeomorphism]]

S qnD qn/D qn S^{q-n} \simeq D^{q-n}/\partial D^{q-n}

which is a [[diffeomorphism]] away from the basepoint and hence such that the action of the [[orthogonal group]] O(qn)O(q-n) induces a smooth action on D qnD^{q-n} (which happens to be constant on D qn\partial D^{q-n}).

Also observe that we may think of the above quotient by the group action

(x,gy)(xg,y) (x, g y) \simeq (x g , y)

as being the quotient by the diagonal action

O(qn)×(O(q) +S qn)(O(q) +S qn) O(q-n) \times ( O(q)_+ \wedge S^{q-n} ) \longrightarrow (O(q)_+ \wedge S^{q-n})

given by

(g,(x,y))(xg 1,gy). (g, (x,y)) \mapsto (x g^{-1}, g y) \,.

Using this we may rewrite the space in question as

(O(q) + O(qn)S qn) (O(q) +S qn)/O(qn) O(q)×D qnO(q)×D qn/O(qn) O(q)×D qn(O(q)×D qn)/O(qn) (O(q)×D qn)/O(qn)((O(q)×D qn))/O(qn). \begin{aligned} (O(q)_+ \wedge_{O(q-n)} S^{q-n} ) & \simeq ( O(q)_+ \wedge S^{q-n} )/ O(q-n) \\ &\simeq \frac{ O(q) \times D^{q-n} }{ O(q) \times \partial D^{q-n} } / O(q-n) \\ & \simeq \frac{ O(q) \times D^{q-n} }{ \partial( O(q) \times D^{q-n} ) } / O(q-n) \\ & \simeq \frac{ (O(q) \times D^{q-n})/ O(q-n) }{ (\partial(O(q)\times D^{q-n}))/O(q-n) } \end{aligned} \,.

Here O(q)×D qnO(q)\times D^{q-n} has the structure of a [[smooth manifold]] [[manifold with boundary|with boundary]] and equipped with a smooth [[action]] of the [[compact Lie group]] O(qn)O(q-n). Under these conditions (Illman 83, corollary 7.2) states that O(q)×D qnO(q) \times D^{q-n} admits the structure of a [[G-CW complex]] for G=O(qn)G = O(q-n), and moreover (Illman 83, line above theorem 7.1) states that this may be chosen such that the boundary is a GG-CW subcomplex.

Now the quotient of a GG-CW complex by GG is a [[CW complex]], and so the last expression above exhibits the quotient of a CW-complex by a subcomplex, hence exhibits CW-complex structure.

Proposition

Let Dia{Top cg,fin */,Orth,Sym}Dia \in \{Top^{\ast/}_{cg,fin}, Orth, Sym\} be the diagram shape of either [[pre-excisive functors]], [[orthogonal spectra]] or [[symmetric spectra]]. Then under the [[symmetric monoidal smash product of spectra]] (def. , def. , def.) the [[free structured spectra]] of def. behave as follows

F n 1 dia(K 1) 𝕊 diaF n 2 dia(K 2)F n 1+n 2(K 1K 2). F^{dia}_{n_1}(K_1) \otimes_{\mathbb{S}_{dia}} F^{dia}_{n_2}(K_2) \;\simeq\; F_{n_1 + n_2}(K_1 \wedge K_2) \,.

In particular for structured [[suspension spectra]] Σ dia F 0 dia\Sigma^\infty_{dia}\coloneqq F_0^{dia} (def. ) this gives isomorphisms

Σ dia (K 1) 𝕊 diaΣ dia (K 2)Σ dia (K 1K 2). \Sigma^\infty_{dia}(K_1) \otimes_{\mathbb{S}_{dia}} \Sigma^\infty_{dia}(K_2) \;\simeq\; \Sigma^\infty_{dia}(K_1 \wedge K_2) \,.

Hence the structured [[suspension spectrum]] functor Σ dia \Sigma^\infty_{dia} is a [[strong monoidal functor]] (def. ) and in fact a [[braided monoidal functor]] (def. ) from [[pointed topological spaces]] equipped with the [[smash product]] of pointed objects, to [[structured spectra]] equipped with the [[symmetric monoidal smash product of spectra]]

Σ dia :(Top cg */,,S 0)(𝕊 diaMod, 𝕊 dia,𝕊 dia). \Sigma_{dia}^\infty \;\colon\; (Top^{\ast/}_{cg},\wedge, S^0) \longrightarrow ( \mathbb{S}_{dia}Mod, \otimes_{\mathbb{S}_{dia}}, \mathbb{S}_{dia} ) \,.

More generally, for X𝕊 diaModX \in \mathbb{S}_{dia}Mod then

X 𝕊 dia(Σ dia K)XK, X \otimes_{\mathbb{S}_{dia}} ( \Sigma^\infty_{dia} K ) \simeq X \wedge K \,,

where on the right we have the smash tensoring of XX with KTop cg */K \in Top^{\ast/}_{cg}.

(MMSS 00, lemma 1.8 with theorem 2.2, Mandell-May 02, prop. 2.2.6)

Proof

By lemma the free spectra are [[free modules]] over the structured [[sphere spectrum]] 𝕊 dia\mathbb{S}_{dia} of the form F n dia(K)𝕊 dia Day(y(n)K)F^{dia}_n(K) \simeq \mathbb{S}_{dia} \otimes_{Day} ( y(n) \wedge K ). By example the tensor product of such free modules is given by

(𝕊 dia Day(y(n 1)K 1)) Day(𝕊 dia Day(y(n 2)K 2))𝕊 dia Day(y(n 1)K) Day(y(n 2)K). \left( \mathbb{S}_{dia} \otimes_{Day} (y(n_1) \wedge K_1) \right) \otimes_{Day} \left( \mathbb{S}_{dia} \otimes_{Day} ( y(n_2) \wedge K_2 ) \right) \;\simeq\; \mathbb{S}_{dia} \otimes_{Day} ( y(n_1) \wedge K ) \otimes_{Day} ( y(n_2) \wedge K ) \,.

Using the [[co-Yoneda lemma]] (prop. ) the expression on the right is

((y(n 1)K 1) Day(y(n 2)K 2))(c) =c 1, 2Dia(c 1+c 2,c)y(n 1)(c 1)K 1y(n 2)(c 2)K 2 c 1,c 2Dia(c 1+c 2,c)Dia(n 1,c 1)Dia(n 2,c 2)K 1K 2 Dia(n 1+n 2,c)K 1K 2 (y(n 1+n 2)(K 1K 2))(c). \begin{aligned} \left( (y(n_1) \wedge K_1) \otimes_{Day} (y(n_2) \wedge K_2) \right)(c) & = \overset{c_1,_2}{\int} Dia(c_1 + c_2, c) \wedge y(n_1)(c_1) \wedge K_1 \wedge y(n_2)(c_2) \wedge K_2 \\ & \simeq \overset{c_1,c_2}{\int} Dia(c_1 + c_2, c) \wedge Dia(n_1,c_1) \wedge Dia(n_2,c_2) \wedge K_1 \wedge K_2 \\ & \simeq Dia(n_1 + n_2,c) \wedge K_1 \wedge K_2 \\ & \simeq \left(y(n_1 + n_2) \wedge (K_1 \wedge K_2)\right)(c) \end{aligned} \,.

For the last statement we may use that Σ dia K𝕊 diaK\Sigma^\infty_{dia} K \simeq \mathbb{S}_{dia} \wedge K, by lemma ), and that 𝕊 dia\mathbb{S}_{dia} is the [[tensor unit]] for 𝕊 dia\otimes_{\mathbb{S}_{dia}} by prop. .

To see that Σ dia \Sigma^\infty_{dia} is braided, write Σ dia K𝕊K\Sigma^\infty_{dia}K\simeq \mathbb{S} \wedge K. We need to see that

(𝕊K 1) 𝕊(𝕊K 2) (𝕊K 2) 𝕊(𝕊K 1) 𝕊(K 1K 2) 𝕊(K 2K 1) \array{ (\mathbb{S} \wedge K_1) \otimes_{\mathbb{S}} (\mathbb{S} \wedge K_2) &\overset{}{\longrightarrow}& (\mathbb{S} \wedge K_2) \otimes_{\mathbb{S}} (\mathbb{S} \wedge K_1) \\ \downarrow && \downarrow \\ \mathbb{S} \wedge (K_1 \wedge K_2) &\underset{}{\longrightarrow}& \mathbb{S} \wedge (K_2 \wedge K_1) }

commutes. Chasing the smash factors through this diagram and using symmetry (def. ) and the hexagon identities (def. ) shows that indeed it does.

One use of free spectra is that they serve to co-represent adjuncts of structure morphisms of spectra. To this end, first consider the following general existence statement.

Lemma

For each nn \in \mathbb{N} there exists a morphism

λ n:F n+1 diaS 1F n diaS 0 \lambda_n \;\colon\; F_{n+1}^{dia}S^1 \longrightarrow F_n^{dia} S^0

between [[free spectra]] (def. ) such that for every structured spectrum X𝕊 diaModX\in \mathbb{S}_{dia} Mod precomposition λ n *\lambda_n^\ast forms a [[commuting diagram]] of the form

𝕊 diaMod(F n diaS 0,X) Top */(S 0,X n) X n λ n * σ˜ n X 𝕊 diaMod(F n+1 diaS 1,X) Top */(S 1,X n+1) ΩX n+1, \array{ \mathbb{S}_{dia}Mod(F^{dia}_n S^0, X) &\simeq& Top^{\ast/}(S^0,X_n) &\simeq& X_n \\ \downarrow^{\mathrlap{\lambda_n^\ast}} && && \downarrow^{\mathrlap{\tilde \sigma^X_n}} \\ \mathbb{S}_{dia}Mod(F^{dia}_{n+1} S^1, X) &\simeq& Top^{\ast/}(S^1, X_{n+1}) &\simeq& \Omega X_{n+1} } \,,

where the horizontal equivalences are the [[adjunction]] isomorphisms and the canonical identification, and where the right morphism is the (ΣΩ)(\Sigma \dashv \Omega)-[[adjunct]] of the structure map σ n\sigma_n of the [[sequential spectrum]] seq *Xseq^\ast X underlying XX (def. ).

Proof

Since all prescribed morphisms in the diagram are [[natural transformations]], this is in fact a diagram of [[copresheaves]] on 𝕊 diaMod\mathbb{S}_{dia} Mod

𝕊 diaMod(F n diaS 0,) Top */(S 0,() n) () n σ˜ n () 𝕊 diaMod(F n+1 diaS 1,) Top */(S 1,() n+1) Ω() n+1. \array{ \mathbb{S}_{dia}Mod(F^{dia}_n S^0, -) &\simeq& Top^{\ast/}(S^0,(-)_n) &\simeq& (-)_n \\ \downarrow^{\mathrlap{}} && && \downarrow^{\mathrlap{\tilde \sigma^{(-)}_n}} \\ \mathbb{S}_{dia}Mod(F^{dia}_{n+1} S^1, -) &\simeq& Top^{\ast/}(S^1, (-)_{n+1}) &\simeq& \Omega (-)_{n+1} } \,.

With this the statement follows by the [[Yoneda lemma]].

Now we say explicitly what these maps are:

Definition

For nn \in \mathbb{N}, write

λ n:F n+1S 1F nS 0 \lambda_n \;\colon\; F_{n+1} S^1 \longrightarrow F_n S^0

for the [[adjunct]] under the ([[free structured spectrum]] \dashv nn-component)-[[adjunction]] in def. of the composite morphism

S 1=(F n Seq(S 0)) n+1(f n Seq) n+1(F n diaS 0) n+1, S^1 \stackrel{=}{\to} (F_n^{Seq}(S^0))_{n+1} \stackrel{(f_n^{Seq})_{n+1}}{\hookrightarrow} (F^{dia}_n S^0)_{n+1} \,,

where the first morphism is via prop. and the second comes from the adjunction units according to def. .

(MMSS 00, def. 8.4, Schwede 12, example 4.26)

Lemma

The morphisms of def. are those whose existence is asserted by prop. .

(MMSS 00, lemma 8.5, following Hovey-Shipley-Smith 00, remark 2.2.12)

Proof

Consider the case Dia=SeqDia = Seq and n=0n = 0. All other cases work analogously.

By lemma , in this case the morphism λ 0\lambda_0 has components like so:

S 3 id S 3 S 2 id S 2 S 1 id S 1 * 0 S 0 F 1S 1 λ 0 F 0S 0. \array{ \vdots && \vdots \\ S^3 &\stackrel{id}{\longrightarrow}& S^3 \\ S^2 &\stackrel{id}{\longrightarrow}& S^2 \\ S^1 &\stackrel{id}{\longrightarrow}& S^1 \\ \ast &\stackrel{0}{\longrightarrow}& S^0 \\ \underbrace{\,\,\,} && \underbrace{\,\,\,} \\ F_1 S^1 &\stackrel{\lambda_0}{\longrightarrow}& F_0 S^0 } \,.

Now for XX any sequential spectrum, then a morphism f:F 0S 0Xf \colon F_0 S^0 \to X is uniquely determined by its 0th components f 0:S 0X 0f_0 \colon S^0 \to X_0 (that’s of course the very free property of F 0S 0F_0 S^0); as the compatibility with the structure maps forces the first component, in particular, to be σ 0 XΣf\sigma_0^X\circ \Sigma f:

ΣS 0 Σf ΣX 0 σ 0 X S 1 σ 0 XΣf X 1 \array{ \Sigma S^0 &\stackrel{\Sigma f}{\longrightarrow}& \Sigma X_0 \\ \downarrow^{\mathrlap{\simeq}} && \downarrow^{\mathrlap{\sigma_0^X}} \\ S^1 &\stackrel{\sigma_0^X \circ \Sigma f}{\longrightarrow}& X_1 }

But that first component is just the component that similarly determines the precompositon of ff with λ 0\lambda_0, hence λ 0 *f\lambda_0^\ast f is fully fixed as being the map σ 0 XΣf\sigma_0^X \circ \Sigma f. Therefore λ 0 *\lambda_0^\ast is the function

λ 0 *:X 0=Maps(S 0,X 0)fσ 0 XΣfMaps(S 1,X 1)=ΩX 1. \lambda_0^\ast \;\colon\; X_0 = Maps(S^0, X_0) \stackrel{f \mapsto \sigma_0^X \circ \Sigma f}{\longrightarrow} \Maps(S^1, X_1) = \Omega X_1 \,.

It remains to see that this is the (ΣΩ)(\Sigma \dashv \Omega)-[[adjunct]] of σ 0 X\sigma_0^X. By the general formula for adjuncts, this is

σ˜ 0 X:X 0ηΩΣX 0Ωσ 0 XΩX 1. \tilde \sigma_0^X \;\colon\; X_0 \stackrel{\eta}{\longrightarrow} \Omega \Sigma X_0 \stackrel{\Omega \sigma_0^X}{\longrightarrow} \Omega X_1 \,.

To compare to the above, we check what this does on points: S 0f 0X 0S^0 \stackrel{f_0}{\longrightarrow} X_0 is sent to the composite

S 0f 0X 0ηΩΣX 0Ωσ 0 XΩX 1. S^0 \stackrel{f_0}{\longrightarrow} X_0 \stackrel{\eta}{\longrightarrow} \Omega \Sigma X_0 \stackrel{\Omega \sigma_0^X}{\longrightarrow} \Omega X_1 \,.

To identify this as a map S 1X 1S^1 \to X_1 we use the adjunction isomorphism once more to throw all the Ω\Omega-s on the right back to Σ\Sigma-s the left, to finally find that this is indeed

σ 0 XΣf:S 1=ΣS 0ΣfΣX 0σ 0 XX 1. \sigma_0^X \circ \Sigma f \;\colon\; S^1 = \Sigma S^0 \stackrel{\Sigma f}{\longrightarrow} \Sigma X_0 \stackrel{\sigma_0^X}{\longrightarrow} X_1 \,.
Lemma

The maps λ n:F n+1S 1F nS 0\lambda_n \;\colon\; F_{n+1} S^1 \longrightarrow F_n S^0 in def. are

  1. [[stable weak homotopy equivalences]] for [[sequential spectra]], [[orthogonal spectra]] and [[pre-excisive functors]], i.e. for Dia{Top */,Orth,Seq}{Dia} \in \{Top^{\ast/}, Orth, Seq\};

  2. not stable weak homotopy equivalences for the case of symmetric spectra Dia=Sym{Dia} = {Sym}.

(Hovey-Shipley-Smith 00, example 3.1.10, MMSS 00, lemma 8.6, Schwede 12, example 4.26)

Proof

This follows by inspection of the explicit form of the maps, via prop. . We discuss each case separately:

sequential case

Here the components of the morphism eventually stabilize to isomorphisms

(λ n) n+3 S 3 id S 3 (λ n) n+2 S 2 id S 2 (λ n) n+1 S 1 id S 1 (λ n) n: * 0 S 0 * * * * λ n: F n+1S 1 F nS 0 \array{ & \vdots && \vdots \\ (\lambda_n)_{n+3} & S^3 &\stackrel{id}{\longrightarrow}& S^3 \\ (\lambda_n)_{n+2} & S^2 &\stackrel{id}{\longrightarrow}& S^2 \\ (\lambda_n)_{n+1} & S^1 &\stackrel{id}{\longrightarrow}& S^1 \\ (\lambda_n)_n \colon & \ast &\stackrel{0}{\longrightarrow}& S^0 \\ & \ast &\longrightarrow& \ast \\ & \vdots && \vdots \\ & \ast &\longrightarrow& \ast \\ & \underbrace{\,\,\,} && \underbrace{\,\,\,} \\ \lambda_n \colon & F_{n+1} S^1 &\stackrel{}{\longrightarrow}& F_n S^0 }

and this immediately gives that λ n\lambda_n is an isomorphism on [[stable homotopy groups]].

orthogonal case

Here for qn+1q \geq n+1 the qq-component of λ n\lambda_n is the [[quotient]] map

(λ n) q:O(q) + O(qn1)S qnO(q) + O(qn1)S 1S qn1O(q) + O(qn)S qn. (\lambda_n)_q \;\colon\; O(q)_+ \wedge_{O(q-n-1)} S^{q-n} \simeq O(q)_+ \wedge_{O(q-n-1)} S^1 \wedge S^{q-n-1} \longrightarrow O(q)_+ \wedge_{O(q-n)}S^{q-n} \,.

By the suspension isomorphism for [[stable homotopy groups]], λ n\lambda_n is a stable weak homotopy equivalence precisely if any of its [[reduced suspension|suspensions]] is. Hence consider instead Σ nλ nS nλ n\Sigma^n \lambda_n \coloneqq S^n \wedge \lambda_n, whose qq-component is

(Σ nλ n) q:O(q) + O(qn1)S qO(q) + O(qn)S q. (\Sigma^n\lambda_n)_q \;\colon\; O(q)_+ \wedge_{O(q-n-1)} S^{q} \longrightarrow O(q)_+ \wedge_{O(q-n)}S^{q} \,.

Now due to the fact that O(qk)O(q-k)-action on S qS^q lifts to an O(q)O(q)-action, the quotients of the diagonal action of O(qk)O(q-k) equivalently become quotients of just the left action. Formally this is due to the existence of the [[commuting diagram]]

O(q) +S q id O(q) +S q id O(q) +S q p 2 Q(q) + Q(qk)S q Q(q) + Q(q)S q S q \array{ O(q)_+ \wedge S^q &\stackrel{id}{\longrightarrow}& O(q)_+ \wedge S^q &\stackrel{id}{\longrightarrow}& O(q)_+ \wedge S^q \\ \downarrow && \downarrow && \downarrow^{\mathrlap{p_2}} \\ Q(q)_+ \wedge_{Q(q-k)} S^q &\longrightarrow& Q(q)_+ \wedge_{Q(q)} S^q & \stackrel{\simeq}{\longrightarrow} & S^q }

which says that the image of any (g,s)O(q) +S q(g,s) \in O(q)_+ \wedge S^q in the quotient Q(q) + Q(qk)S qQ(q)_+ \wedge_{Q(q-k)} S^q is labeled by ([g],s)([g],s).

It follows that (Σ nλ n) q(\Sigma^n\lambda_n)_q is the smash product of a projection map of [[coset spaces]] with the identity on the sphere:

(Σ nλ n) qproj +id S q:O(q)/O(qn1) +S qO(q)/O(qn) +S q. (\Sigma^n\lambda_n)_q \simeq proj_+ \wedge id_{S^q} \;\colon\; O(q)/O(q-n-1)_+ \wedge S^q \longrightarrow O(q)/O(q-n)_+ \wedge S^{q} \,.

Now finally observe that this projection function

proj:O(q)/O(qn1)O(q)/O(qn) proj \;\colon\; O(q)/O(q-n-1) \longrightarrow O(q)/O(q-n)

is (qn1)(q - n -1 )-connected (see here). Hence its smash product with S qS^q is (2qn1)(2q - n -1 )-connected.

The key here is the fast growth of the connectivity with qq. This implies that for each ss there exists qq such that π s+q((Σ nλ n) q)\pi_{s+q}((\Sigma^n \lambda_n)_q) becomes an isomorphism. Hence Σ nλ n\Sigma^n \lambda_n is a stable weak homotopy equivalence and therefore so is λ n\lambda_n.

symmetric case

Here the morphism λ n\lambda_n has the same form as in the orthogonal case above, except that all occurences of [[orthogonal groups]] are replaced by just their sub-[[symmetric groups]].

Accordingly, the analysis then proceeds entirely analogously, with the key difference that the projection

Σ(q)/Σ(qn1)Σ(q)/Σ(qn) \Sigma(q)/\Sigma(q-n-1) \longrightarrow \Sigma(q)/\Sigma(q-n)

does not become highly connected as qq increases, due to the [[discrete topological space]] underlying the symmetric group. Accordingly the conclusion now is the opposite: λ n\lambda_n is not a stable weak homotopy equivalence in this case.

Another use of free spectra is that their [[pushout products]] may be explicitly analyzed, and checking the [[pushout-product axiom]] for general cofibrations may be reduced to checking it on morphisms between free spectra.

Lemma

The [[symmetric monoidal smash product of spectra]] of the [[free spectrum]] constructions (def. ) on the generating cofibrations {S n1i nD n} n𝔹\{S^{n-1}\overset{i_n}{\hookrightarrow} D^n\}_{n \in \mathbb{B}} of the [[classical model structure on topological spaces]] is given by addition of indices

(F ki n 1) 𝕊 dia(F i n 2)F k+(i n 1+n 2). (F_k i_{n_1}) \Box_{\mathbb{S}_{dia}} (F_\ell i_{n_2}) \simeq F_{k+\ell}( i_{n_1 + n_2}) \,.
Proof

By lemma the [[commuting diagram]] defining the [[pushout product]] of [[free spectra]]

F kS + n 11 𝕊 diaF S + n 21 F kD + n 1 𝕊 diaF S + n 21 F kS + n 11 𝕊 diaF D + n 21 F kD + n 11 𝕊 diaF kD + n 21 \array{ && F_k S^{n_1-1}_+ \wedge_{\mathbb{S}_{dia}} F_{\ell} S^{n_2-1}_+ \\ & \swarrow && \searrow \\ F_k D^{n_1}_+ \wedge_{\mathbb{S}_{dia}} F_{\ell} S^{n_2-1}_+ && && F_k S^{n_1-1}_+ \wedge_{\mathbb{S}_{dia}} F_{\ell} D^{n_2-1}_+ \\ & \searrow && \swarrow \\ && F_k D^{n_1-1}_+ \wedge_{\mathbb{S}_{dia}} F_k D^{n_2-1}_+ }

is equivalent to this diagram:

F k+((S n 11×S n 21) +) F k+((D n 1×S n 21) +) F k+((S n 11×D n 2) +) F k+((D n 1×D n 2) +). \array{ && F_{k+\ell}((S^{n_1-1}\times S^{n_2-1})_+) \\ & \swarrow && \searrow \\ F_{k+\ell}((D^{n_1} \times S^{n_2-1})_+) && && F_{k+\ell}((S^{n_1-1} \times D^{n_2})_+) \\ & \searrow && \swarrow \\ && F_{k+ \ell}( (D^{n_1}\times D^{n_2})_+ ) } \,.

Since the [[free spectrum]] construction is a left adjoint, it preserves pushouts, and so

(F ki n 1) 𝕊 dia(F i n 2)F k+(i n 1i n 2)F k+(i n 1+n 2), (F_{k}i_{n_1}) \Box_{\mathbb{S}_{dia}} (F_{\ell}i_{n_2}) \simeq F_{k + \ell}( i_{n_1} \Box i_{n_2}) \simeq F_{k + \ell}( i_{n_1 + n_2}) \,,

where in the second step we used this lemma.

The strict model structure on structured spectra

Theorem

The four categories of

  1. [[pre-excisive functors]] Exc(Top cg)Exc(Top_{cg});

  2. [[orthogonal spectra]] OrthSpec(Top cg)=𝕊 orthModOrthSpec(Top_{cg}) = \mathbb{S}_{orth} Mod;

  3. [[symmetric spectra]] SymSpec(Top cg)=𝕊 symModSymSpec(Top_{cg}) = \mathbb{S}_{sym}Mod;

  4. [[sequential spectra]] SeqSpec(Top cg)=𝕊 seqModSeqSpec(Top_{cg}) = \mathbb{S}_{seq}Mod

(from def. , prop. , def. ) each admit a [[model category]] structure (def.) whose weak equivalences and fibrations are those morphisms which induce on all component spaces weak equivalences or fibrations, respectively, in the [[classical model structure on pointed topological spaces]] (Top cg */) Quillen(Top^{\ast/}_{cg})_{Quillen}. (thm., prop.). These are called the strict model structures (or level model structures) on [[structured spectra]].

Moreover, under the [[equivalences of categories]] of prop. and prop. , the restriction functors in def. constitute [[right adjoints]] of [[Quillen adjunctions]] (def.) between these model structures:

Exc(Top cg) strict OrthSpec(Top cg) strict SymSpec(Top cg) strict SeqSpec(Top cg) strict 𝕊Mod strict orth *orth ! 𝕊 OrthMod strict sym *sym ! 𝕊 SymMod strict seq *seq ! 𝕊 SeqMod strict. \array{ Exc(Top_{cg})_{strict} && OrthSpec(Top_{cg})_{strict} && SymSpec(Top_{cg})_{strict} && SeqSpec(Top_{cg})_{strict} \\ \downarrow^{\mathrlap{\simeq}} && \downarrow^{\mathrlap{\simeq}} && \downarrow^{\mathrlap{\simeq}} && \downarrow^{\mathrlap{\simeq}} \\ \mathbb{S} Mod_{strict} & \underoverset {\underset{orth^\ast}{\longrightarrow}} {\overset{orth_!}{\longleftarrow}} {\bot} & \mathbb{S}_{Orth} Mod_{strict} & \underoverset {\underset{sym^\ast}{\longrightarrow}} {\overset{sym_!}{\longleftarrow}} {\bot} & \mathbb{S}_{Sym} Mod_{strict} & \underoverset {\underset{seq^\ast}{\longrightarrow}} {\overset{seq_!}{\longleftarrow}} {\bot} & \mathbb{S}_{Seq} Mod_{strict} } \,.

(MMSS 00, theorem 6.5)

Proof

By prop. all four categories are equivalently categories of pointed [[topologically enriched functors]]

𝕊 diaMod[𝕊 diaFree diaMod,Top cg */] \mathbb{S}_{dia}Mod \simeq [ \mathbb{S}_{dia} Free_{dia}Mod, Top^{\ast/}_{cg} ]

and hence the existence of the model structures with componentwise weak equivalences and fibrations is a special case of the general existence of the [[projective model structure on enriched functors]] (thm.).

The three restriction functors dia *dia^\ast each have a [[left adjoint]] dia !dia_! by topological [[left Kan extension]] (prop. ).

Moreover, the three right adjoint restriction functors are along inclusions of objects, hence evidently preserve componentwise weak equivalences and fibrations. Hence these are [[Quillen adjunctions]].

Definition

Recall the sets

I Top */{S + n1(i n) +D + n} n I_{Top^{\ast/}} \coloneqq \{S^{n-1}_+ \overset{(i_n)_+}{\hookrightarrow} D^n_+\}_{n \in \mathbb{N}}
J Top */{D + n(j n) +(D n×I) +} n J_{Top^{\ast/}} \coloneqq \{D^n_+ \overset{(j_n)_+}{\hookrightarrow} (D^n \times I)_+\}_{n \in \mathbb{N}}

of generating cofibrations and generating acyclic cofibrations, respectively, of the [[classical model structure on pointed topological spaces]] (def.)

Write

I dia strict{F c dia((i n) +)} cDia,n I^{strict}_{dia} \;\coloneqq\; \left\{ F_c^{dia}((i_n)_+) \right\}_{c \in Dia, n \in \mathbb{N}}

for the set of images under forming [[free spectra]], def. , on the morphisms in I Top */I_{Top^{\ast/}} from above. Similarly, write

J dia strict{F c dia((j n) +)}, J^{strict}_{dia} \;\coloneqq\; \left\{ F_c^{dia}((j_n)_+) \right\} \,,

for the set of images under forming free spectra of the morphisms in J Top cg */J_{Top^{\ast/}_{cg}}.

Proposition

The sets I dia strictI^{strict}_{dia} and J dia strictJ^{strict}_{dia} from def. are, respectively sets of [[generating cofibrations]] and generating acyclic cofibrations that exhibit the strict model structure 𝕊 DiaMod strict\mathbb{S}_{Dia}Mod_{strict} from theorem as a [[cofibrantly generated model category]] (def.).

(MMSS 00, theorem 6.5)

Proof

By theorem the strict model structure is equivalently the projective pointed [[model structure on enriched functors|model structure on topologically enriched functors]]

𝕊 DiaMod strict[𝕊 DiaFree DiaMod op,Top */] proj \mathbb{S}_{Dia}Mod_{strict} \simeq [\mathbb{S}_{Dia}Free_{Dia}Mod^{op}, Top^{\ast/}]_{proj}

of the opposite of the category of free spectra on objects in 𝒞[𝒞,Top cg */]\mathcal{C} \hookrightarrow [\mathcal{C}, Top^{\ast/}_{cg}].

By the general discussion in [[Introduction to Stable homotopy theory – P|Part P – Classical homotopy theory]] (this theorem) the [[projective model structure on functors]] is cofibrantly generated by the smash tensoring of the [[representable functors]] with the elements in I Top cg */I_{Top^{\ast/}_{cg}} and J Top cg */J_{Top^{\ast/}_{cg}}. By the proof of lemma , these are precisely the morphisms of free spectra in I dia strictI^{strict}_{dia} and J dia strictJ^{strict}_{dia}, respectively.

Topological enrichment

By the general properties of the [[projective model structure on functors|projective model structure]] on [[topologically enriched functors]], theorem implies that the strict model category of structured spectra inherits the structure of an [[enriched model category]], enriched over the [[classical model structure on pointed topological spaces]]. This proceeds verbatim as for sequential spectra (in part 1.1 – Topological enrichement), but for ease of reference we here make it explicit again.

Definition

Let Dia{Top cg,fin */,Orth,Sym,Seq}Dia \in \{Top^{\ast/}_{cg,fin}, Orth, Sym, Seq\} one of the shapes for structured spectra from def. .

Let f:XYf \;\colon \; X \to Y be a morphism in 𝕊 diaMod\mathbb{S}_{dia}Mod (as in prop. ) and let i:ABi \;\colon\; A \to B a morphism in Top cg */Top_{cg}^{\ast/}.

Their [[pushout product]] with respect to smash tensoring is the universal morphism

fi((id,i),(f,id)) f \Box i \coloneqq \left((id,i), (f,id)\right)

in

XA (f,id) (id,i) YA (po) XB (YA)XA(XB) ((id,i),(f,id)) YB, \array{ && X \wedge A \\ & {}^{\mathllap{(f,id)}}\swarrow && \searrow^{\mathrlap{(id,i)}} \\ Y \wedge A && (po) && X \wedge B \\ & {}_{\mathllap{}}\searrow && \swarrow \\ && (Y \wedge A) \underset{X \wedge A}{\sqcup} (X \wedge B) \\ && \downarrow^{\mathrlap{((id, i), (f,id))}} \\ && Y \wedge B } \,,

where

()():𝕊 diaMod×Top cg */[𝕊 diaFre diaMod op,Top cg */]×Top cg */[𝕊 diaFre diaMod op,Top cg */]𝕊 diaMod (-)\wedge(-) \;\colon\; \mathbb{S}_{dia}Mod \times Top^{\ast/}_{cg} \simeq [ \mathbb{S}_{dia}Fre_{dia}Mod^{op},\; Top^{\ast/}_{cg}] \times Top^{\ast/}_{cg} \longrightarrow [ \mathbb{S}_{dia}Fre_{dia}Mod^{op},\; Top^{\ast/}_{cg}] \simeq \mathbb{S}_{dia}Mod

denotes the smash tensoring of pointed topologically enriched functors with pointed topological spaces (def.)

Dually, their pullback powering is the universal morphism

f i(Maps(B,f) *,Maps(i,X) *) f^{\Box i} \coloneqq (Maps(B,f)_\ast, Maps(i,X)_\ast)

in

Maps(B,X) * (Maps(B,f) *,Maps(i,X) *) Maps(B,Y) *×Maps(A,Y) *Maps(A,X) * Maps(B,Y) * (pb) Maps(A,X) * Maps(i,Y) * Maps(A,p) * Maps(A,Y) *, \array{ && Maps(B,X)_\ast \\ && \downarrow^{\mathrlap{(Maps(B,f)_\ast, Maps(i,X)_\ast)}} \\ && Maps(B,Y)_\ast \underset{Maps(A,Y)_\ast}{\times} Maps(A,X)_\ast \\ & \swarrow && \searrow \\ Maps(B,Y)_\ast && (pb) && Maps(A,X)_\ast \\ & {}_{\mathllap{Maps(i,Y)_\ast}}\searrow && \swarrow_{\mathrlap{Maps(A,p)_\ast}} \\ && Maps(A,Y)_\ast } \,,

where

Maps(,) *:(Top cg *) op×𝕊 diaMod(Top cg */) op×[𝕊 diaFree DiaMod op,Top cg */][𝕊 diaFree DiaMod op,Top cg */]𝕊 diaMod Maps(-,-)_\ast \;\colon\; (Top^{\ast}_{cg})^{op} \times \mathbb{S}_{dia}Mod \simeq (Top^{\ast/}_{cg})^{op} \times [\mathbb{S}_{dia}Free_{Dia}Mod^{op},Top^{\ast/}_{cg}] \longrightarrow [\mathbb{S}_{dia}Free_{Dia}Mod^{op},Top^{\ast/}_{cg}] \simeq \mathbb{S}_{dia}Mod

denotes the smash powering (def.).

Finally, for f:XYf \colon X \to Y and i:ABi \colon A \to B both morphisms in 𝕊 diaMod\mathbb{S}_{dia}Mod, then their pullback powering is the universal morphism

f i(𝕊 diaMod(B,f),𝕊 diaMod(i,X)) f^{\Box i} \coloneqq (\mathbb{S}_{dia}Mod(B,f), \mathbb{S}_{dia}Mod(i,X))

in

𝕊 diaMod(B,X) (𝕊 diaMod(B,f),𝕊 diaMod(i,X)) 𝕊 diaMod(B,Y)×𝕊 diaMod(A,Y)𝕊 diaMod(A,X) 𝕊 diaMod(B,Y) (pb) 𝕊 diaMod(A,X) 𝕊 diaMod(i,Y) 𝕊 diaMod(A,p) 𝕊 diaMod(A,Y), \array{ && \mathbb{S}_{dia}Mod(B,X) \\ && \downarrow^{\mathrlap{(\mathbb{S}_{dia}Mod(B,f), \mathbb{S}_{dia}Mod(i,X))}} \\ && \mathbb{S}_{dia}Mod(B,Y) \underset{\mathbb{S}_{dia}Mod(A,Y)}{\times} \mathbb{S}_{dia}Mod(A,X) \\ & \swarrow && \searrow \\ \mathbb{S}_{dia}Mod(B,Y) && (pb) && \mathbb{S}_{dia}Mod(A,X) \\ & {}_{\mathllap{\mathbb{S}_{dia}Mod(i,Y)}}\searrow && \swarrow_{\mathrlap{\mathbb{S}_{dia}Mod(A,p)}} \\ && \mathbb{S}_{dia}Mod(A,Y) } \,,

where now 𝕊 diaMod(,)\mathbb{S}_{dia}Mod(-,-) is the [[hom-space]] functor of 𝕊 diaMod[𝕊 diaFree DiaMod op,Top cg */]\mathbb{S}_{dia}Mod \simeq [\mathbb{S}_{dia}Free_{Dia}Mod^{op}, Top^{\ast/}_{cg}] from def. .

Proposition

The operations of forming pushout products and pullback powering with respect to smash tensoring in def. is compatible with the strict model structure 𝕊 diaMod strict\mathbb{S}_{dia}Mod_{strict} on structured spectra from theorem and with the [[classical model structure on pointed topological spaces]] (Top cg */) Quillen(Top^{\ast/}_{cg})_{Quillen} (thm., prop.) in that pushout product takes two cofibrations to a cofibration, and to an acyclic cofibration if at least one of the inputs is acyclic, and pullback powering takes a fibration and a cofibration to a fibration, and to an acylic one if at least one of the inputs is acyclic:

Cof strictCof cl Cof strict Cof strict(Cof clW cl) Cof strictW strict (Cof strictW strict)Cof cl Cof strictW strict. \begin{aligned} Cof_{strict} \Box Cof_{cl} & \subset\; Cof_{strict} \\ Cof_{strict} \Box (Cof_{cl} \Box W_{cl}) & \subset\; Cof_{strict} \cap W_{strict} \\ (Cof_{strict} \cap W_{strict}) \Box Cof_{cl} & \subset\; Cof_{strict} \cap W_{strict} \end{aligned} \,.

Dually, the pullback powering (def. ) satisfies

Fib strict Cof cl Fib strict Fib strict (Cof clW cl) Fib strictW strict (Fib strictW strict) Cof cl Fib strictW strict. \begin{aligned} Fib_{strict}^{\Box Cof_{cl}} & \subset\; Fib_{strict} \\ Fib_{strict}^{\Box ( Cof_{cl} \cap W_{cl})} & \subset\; Fib_{strict}\cap W_{strict} \\ (Fib_{strict} \cap W_{strict})^{\Box Cof_{cl}} & \subset\; Fib_{strict} \cap W_{strict} \end{aligned} \,.
Proof

The statement concering the pullback powering follows directly from the analogous statement for topological spaces (prop.) by the fact that, via theorem , the fibrations and weak equivalences in 𝕊 diaMod strict\mathbb{S}_{dia}Mod_{strict} are degree-wise those in (Top cg */) Quillen(Top_{cg}^{\ast/})_{Quillen}, and since smash tensoring and powering is defined degreewise. From this the statement about the pushout product follows dually by [[Joyal-Tierney calculus]] (prop.).

Remark

In the language of [[model category]]-theory, prop. says that 𝕊 diaMod strict\mathbb{S}_{dia}Mod_{strict} is an [[enriched model category]], the enrichment being over (Top cg */) Quillen(Top_{cg}^{\ast/})_{Quillen}. This is often referred to simply as a “topological model category”.

We record some immediate consequences of prop. that will be useful.

Proposition

Let KTop cg *K \in Top^{\ast}_{cg} be a [[retract]] of a [[cell complex]] (def.), then the smash-tensoring/powering adjunction from prop. is a [[Quillen adjunction]] (def.) for the strict model structure from theorem

𝕊 diaMod(Top cg) strictMaps(K,) *()K𝕊 diaMod(Top cg) strict. \mathbb{S}_{dia}Mod(Top_{cg})_{strict} \underoverset {\underset{Maps(K,-)_\ast}{\longrightarrow}} {\overset{(-)\wedge K}{\longleftarrow}} {\bot} \mathbb{S}_{dia}Mod(Top_{cg})_{strict} \,.
Proof

By assumption, KK is a cofibrant object in the [[classical model structure on pointed topological spaces]] (thm., prop.), hence *K\ast \to K is a cofibration in (Top cg */) Quillen(Top^{\ast/}_{cg})_{Quillen}. Observe then that the the [[pushout product]] of any morphism ff with *K\ast \to K is equivalently the smash tensoring of ff with KK:

f(*K)fK. f \Box (\ast \to K) \simeq f \wedge K \,.

This way prop. implies that ()K(-)\wedge K preserves cofibrations and acyclic cofibrations, hence is a left Quillen functor.

Lemma

Let X𝕊 diaMod strictX \in \mathbb{S}_{dia}Mod_{strict} be a structured spectrum, regarded in the strict model structure of theorem .

  1. The smash powering of XX with the standard topological interval I +I_+ (exmpl.) is a good [[path space object]] (def.)

    Δ X:XW strictX I +Fib strictX×X. \Delta_X \;\colon\; X \overset{\in W_{strict}}{\longrightarrow} X^{I_+} \overset{\in Fib_{strict}}{\longrightarrow} X \times X \,.
  2. If XX is cofibrant, then its smash tensoring with the standard topological interval I +I_+ (exmpl.) is a good [[cylinder object]] (def.)

    X:XXCof strictX(I +)W strictX. \nabla_X \;\colon\; X \vee X \overset{\in Cof_{strict}}{\longrightarrow} X\wedge (I_+) \overset{\in W_{strict}}{\longrightarrow} X \,.
Proof

It is clear that we have weak equivalences as shown (I*I \to \ast is even a [[homotopy equivalence]]), what requires proof is that the path object is indeed good in that X (I +)X×XX^{(I_+)} \to X \times X is a fibration, and the cylinder object is indeed good in that XXX(I +)X \vee X \to X\wedge (I_+) is indeed a cofibration.

For the first statement, notice that the pullback powering (def. ) of **(i 0,i 1)I\ast \sqcup \ast \overset{(i_0,i_1)}{\longrightarrow} I into the terminal morphism X*X \to \ast is the same as the powering X (i 0,i 1)X^{(i_0,i_1)}:

((X*) (i 0,i 1))X (i 0,i 1),. ((X\to\ast)^{\Box(i_0,i_1)}) \;\simeq\; X^{(i_0,i_1)} \,,.

But since every object in 𝕊 diaMod strict\mathbb{S}_{dia}Mod_{strict} is fibrant, so that X*X \to \ast is a fibration, and since (i 0,i 1)(i_0,i_1) is a [[relative cell complex]] inclusion and hence a cofibration in (Top cg */) Quilln(Top^{\ast/}_{cg})_{Quilln}, prop. says that X (i 0,i 1):X I +X×XX^{(i_0,i_1)} \colon X^{I_+}\to X \times X is a fibration.

Dually, observe that

(*X)(i 0,i 1)X(i 0,i 1). (\ast \to X) \Box (i_0, i_1) \;\simeq\; X \wedge (i_0,i_1) \,.

Hence if XX is assumed to be cofibrant, so that *X\ast \to X is a cofibration, then prop. implies that X(i 0,i 1):XXX(I +)X \wedge (i_0,i_1) \colon X \wedge X \to X \wedge (I_+) is a cofibration.

Proposition

For X𝕊 diaModX \in \mathbb{S}_{dia}Mod a [[structured spectrum]], fMor(𝕊 diaMod)f \in Mor(\mathbb{S}_{dia}Mod) any morphism of structured spectra, and for gMor(Top cpt */)g \in Mor(Top_{cpt}^{\ast/}) a morphism of [[pointed topological spaces]], then the [[hom-spaces]] of def. (via prop. ) interact with the pushout-product and pullback-powering from def. in that there is a [[natural isomorphism]]

𝕊 diaMod(fg,X)(𝕊 diaMod(f,X)) g. \mathbb{S}_{dia}Mod(f \Box g, X) \simeq (\mathbb{S}_{dia}Mod(f,X))^{\Box g} \,.
Proof

Since the pointed compactly generated [[mapping space]] functor (exmpl.)

Maps(,) *:(Top cg */) op×Top cg */Top cg */ Maps(-,-)_\ast \;\colon\; \left(Top^{\ast/}_{cg}\right)^{op} \times Top^{\ast/}_{cg} \longrightarrow Top^{\ast/}_{cg}

takes [[colimits]] in the first argument to [[limits]] (cor.) and [[ends]] in the second argument to ends (remark ), and since limits and colimits in 𝕊 diaMod\mathbb{S}_{dia}Mod are computed objectswise (this prop. via prop. ) this follows with the [[end]]-formula for the mapping space (def. ):

𝕊 diaMod(fg,X) =cMaps((fg)(c),X(c)) * cMaps(f(c)g,X(c)) * cMaps(f(c),X(c)) * g (cMaps(f(c),X(c)) *) g (𝕊 diaMod(f,X)) g. \begin{aligned} \mathbb{S}_{dia}Mod(f \Box g, X) & = \underset{c}{\int} Maps( (f \Box g)(c), X(c) )_\ast \\ & \simeq \underset{c}{\int} Maps( f(c) \Box g, X(c) )_\ast \\ & \simeq \underset{c}{\int} Maps( f(c), X(c))_\ast^{\Box g} \\ & \simeq \left( \underset{c}{\int} Maps(f(c), X(c))_\ast \right)^{\Box g} \\ & \simeq (\mathbb{S}_{dia}Mod(f,X))^{\Box g} \end{aligned} \,.
Proposition

For X,Y𝕊 diaMod(Top cg)X,Y \in \mathbb{S}_{dia}Mod(Top_{cg}) two structured spectra with XX cofibrant in the strict model structure of def. , then there is a [[natural bijection]]

π 0𝕊 diaMod(X,Y)[X,Y] strict \pi_0 \mathbb{S}_{dia}Mod(X,Y) \simeq [X,Y]_{strict}

between the [[connected components]] of the [[hom-space]] (def. via prop. ) and the [[hom-set]] in the [[homotopy category of a model category|homotopy category]] (def.) of the strict model structure from theorem .

Proof

By prop. the path components of the [[hom-space]] are the [[left homotopy]] classes of morphisms of structured spectra with respect to the standard [[cylinder spectrum]] X(I +)X \wedge (I_+):

I +SeqSpec(X,Y)X(I +)Y. \frac{ I_+ \longrightarrow SeqSpec(X,Y) }{ X \wedge (I_+) \longrightarrow Y } \,.

Moreover, by lemma the degreewise standard [[reduced cylinder]] X(I +)X \wedge (I_+) of structured spectra is a good [[cylinder object]] on XX in 𝕊 diaMod strict\mathbb{S}_{dia}Mod_{strict}. Hence hom-sets in the strict [[homotopy category of a model category|homotopy category]] out of a cofibrant into a fibrant object are given by standard [[left homotopy]] classes of morphisms

[X,Y] strictHom 𝕊 diaMod(X,Y) / [X,Y]_{strict} \simeq Hom_{\mathbb{S}_{dia}Mod}(X,Y)_{/\sim}

(this lemma). Since XX is cofibrant by assumption and since every object is fibrant in 𝕊 diaMod strict\mathbb{S}_{dia}Mod_{strict}, this is the case. Hence the notion of left homotopy here is that seen by the standard interval, and so the claim follows.

Monoidal model structure

We now combine the concepts of [[model category]] (def.) and [[monoidal category]] (def. ).

Given a category 𝒞\mathcal{C} that is equipped both with the structure of a [[monoidal category]] and of a [[model category]], then one may ask whether these two structures are compatible, in that the [[left derived functor]] (def.) of the [[tensor product]] exists to equip also the [[homotopy category of a model category|homotopy category]] with the structure of a monoidal category. If so, then one may furthermore ask if the [[localization]] functor γ:𝒞Ho(𝒞)\gamma \;\colon\; \mathcal{C} \longrightarrow Ho(\mathcal{C}) is a [[monoidal functor]] (def. ).

The axioms on a [[monoidal model category]] (def. below) are such as to ensure that this is the case.

A key consequence is that, via prop. , for a monoidal model category the localization functor γ\gamma carries monoids to monoids. Applied to the [[stable model category]] of spectra established below, this gives that [[structured ring spectra]] indeed represent [[ring spectra]] in the homotopy category. (In fact much more is true, but requires further proof: there is also a model structure on monoids in the model structure of spectra, and with respect to that the structured ring spectra represent [[A-infinity rings]]/[[E-infinity rings]].)

Definition

A (symmetric) [[monoidal model category]] is a [[model category]] 𝒞\mathcal{C} (def.) equipped with the structure of a [[closed monoidal category|closed]] (def. ) [[symmetric monoidal category|symmetric]] (def. ) [[monoidal category]] (𝒞,,I)(\mathcal{C}, \otimes, I) (def. ) such that the following two compatibility conditions are satisfied

  1. ([[pushout-product axiom]]) For every pair of cofibrations f:XYf \colon X \to Y and f:XYf' \colon X' \to Y', their [[pushout-product]], hence the induced morphism out of the cofibered [[coproduct]] over ways of forming the tensor product of these objects

    f g(XY)XX(YX)YY, f \Box_{\otimes} g \;\coloneqq\; (X \otimes Y') \underset{{X \otimes X'}}{\sqcup} (Y \otimes X') \longrightarrow Y \otimes Y' \,,

    is itself a cofibration, which, furthermore, is acyclic if at least one of ff or ff' is.

    (Equivalently this says that the [[tensor product]] :𝒞×𝒞𝒞\otimes \colon \mathcal{C} \times \mathcal{C} \to \mathcal{C} is a left [[Quillen bifunctor]].)

  2. (unit axiom) For every cofibrant object XX and every cofibrant resolution CofQ1p 1W1\emptyset \overset{\in Cof}{\longrightarrow} Q 1 \underoverset{p_1}{\in W}{\longrightarrow} 1 of the [[tensor unit]] 11, the resulting morphism

    Q1Xp 1X1XIsoWX Q 1 \otimes X \overset{p_1 \otimes X}{\longrightarrow} 1 \otimes X \underoverset{\ell}{\in Iso \subset W}{\longrightarrow} X

is a weak equivalence.

(Hovey 99, def. 4.2.6 Schwede-Shipley 00, def. 3.1, remark 3.2)

Observe some immediate consequences of these axioms:

Remark

Since a [[monoidal model category]] (def. ) is assumed to be [[closed monoidal category|closed monoidal]] (def. ), for every object XX the tensor product X()()XX \otimes (-) \simeq (-) \otimes X is a [[left adjoint]] and hence preserves all [[colimits]]. In particular it preserves the [[initial object]] \emptyset (which is the colimit over the empty diagram).

If follows that the tensor-[[pushout-product axiom]] in def. implies that for XX a cofibrant object, then the functor X()X \otimes (-) preserves cofibrations and acyclic cofibrations, since

f (X)fX. f \Box_\otimes (\emptyset \to X) \simeq f \otimes X \,.

This implies that if the [[tensor unit]] 11 happens to be cofibrant, then the unit axiom in def. is already implied by the pushout-product axiom. This is because then we have a lift in

Q1 Cof W p 1 1 = 1. \array{ \emptyset &\longrightarrow& Q 1 \\ {}^{\mathllap{\in Cof}}\downarrow &\nearrow& \downarrow^{\mathrlap{p_1}}_{\mathrlap{\in W}} \\ 1 &=& 1 } \,.

This lift is a weak equivalence by [[two-out-of-three]] (def.). Since it is hence a weak equivalence between cofibrant objects, it is preserved by the left Quillen functor ()X(-) \otimes X (for any cofibrant XX) by [[Ken Brown’s lemma]] (prop.). Hence now p 1Xp_1 \otimes X is a weak equivalence by [[two-out-of-three]].

Since for all the categories of spectra that we are interested in here the tensor unit is always cofibrant (it is always a version of the [[sphere spectrum]], being the image under the left Quillen functor Σ dia \Sigma^\infty_{dia} of the cofibrant pointed space S 0S^0, prop. ), we may ignore the unit axiom.

Proposition

Let (𝒞,,I)(\mathcal{C}, \otimes, I) be a [[monoidal model category]] (def. ) with cofibrant [[tensor unit]] 11.

Then the [[left derived functor]] L\otimes^L (def.) of the tensor product \otimes exsists and makes the [[homotopy category of a model category|homotopy category]] (def.) into a [[monoidal category]] (Ho(𝒞), L,γ(1))(Ho(\mathcal{C}), \otimes^L, \gamma(1)) (def. ) such that the [[localization]] functor γ:𝒞 cHo(𝒞)\gamma\colon \mathcal{C}_c\to Ho(\mathcal{C}) (thm.) on the [[category of cofibrant objects]] (def.) carries the structure of a [[strong monoidal functor]] (def. )

γ:(𝒞,,1)(Ho(𝒞), L,γ(1)). \gamma \;\colon\; (\mathcal{C}, \otimes, 1) \longrightarrow (Ho(\mathcal{C}), \otimes^L , \gamma(1)) \,.

The first statement is also for instance in (Hovey 99, theorem 4.3.2).

Proof

For the [[left derived functor]] (def.) of the tensor product

𝒞×𝒞𝒞 \otimes \; \mathcal{C}\times \mathcal{C} \longrightarrow \mathcal{C}

to exist, it is sufficient that its restriction to the subcategory

(𝒞×𝒞) c𝒞 c×𝒞 c (\mathcal{C} \times \mathcal{C})_c \simeq \mathcal{C}_c \times \mathcal{C}_c

of cofibrant objects preserves acyclic cofibrations (by [[Ken Brown’s lemma]], here).

Every morphism (f,g)(f,g) in the [[product category]] 𝒞 c×𝒞 c\mathcal{C}_{c}\times \mathcal{C}_{c} (def. ) may be written as a composite of a pairing with an identity morphisms

(f,g):(c 1,d 1)(id c 1,g)(c 1,d 2)(f,id c 2)(c 2,d 2). (f,g) \;\colon\; (c_1, d_1) \overset{(id_{c_1},g)}{\longrightarrow} (c_1,d_2) \overset{(f,id_{c_2})}{\longrightarrow} (c_2,d_2) \,.

Now since the [[pushout product]] (with respect to tensor product) with the initial morphism (*c 1)(\ast \to c_1) is equivalently the tensor product

(*c 1) gid c 1g (\ast \to c_1) \Box_{\otimes} g \;\simeq\; id_{c_1} \otimes g

and

f (*c 2)fid c 2 f \Box_{\otimes} (\ast \to c_2) \;\simeq\; f \otimes id_{c_2}

the [[pushout-product axiom]] (def. ) implies that on the subcategory of cofibrant objects the functor \otimes preserves acyclic cofibrations. (This is why one speaks of a [[Quillen bifunctor]], see also Hovey 99, prop. 4.3.1).

Hence L\otimes^L exists.

By the same decomposition and using the [[universal property]] of the [[localization]] of a category (def.) one finds that for 𝒞\mathcal{C} and 𝒟\mathcal{D} any two [[categories with weak equivalences]] (def.) then the [[localization]] of their [[product category]] is the product category of their localizations:

(𝒞×𝒟)[(W 𝒞×W 𝒟) 1](𝒞[W 𝒞 1])×(𝒟[W 𝒟 1]). (\mathcal{C} \times \mathcal{D})[(W_{\mathcal{C}} \times W_{\mathcal{D}})^{-1}] \simeq (\mathcal{C}[W^{-1}_{\mathcal{C}}]) \times (\mathcal{D}[W^{-1}_{\mathcal{D}}]) \,.

With this, the [[universal property]] as a [[localization]] (def.) of the [[homotopy category of a model category]] (thm.) induces [[associators]] α L\alpha^L and [[unitors]] L\ell^L, r Lr^L on (Ho(𝒞, L))(Ho(\mathcal{C}, \otimes^L )):

First write

μ:γ() Lγ()γ(()()) \mu \;\colon\; \gamma(-) \otimes^L \gamma(-) \overset{\simeq}{\longrightarrow} \gamma( (-) \otimes (-) )

for (the inverse of) the corresponding [[natural isomorphism]] in the localization diagram

𝒞×𝒞 𝒞 γ×γ μ 1 γ Ho(𝒞)×Ho(𝒞) L Ho(𝒞). \array{ \mathcal{C} \times \mathcal{C} &\overset{\otimes}{\longrightarrow}& \mathcal{C} \\ {}^{\mathllap{\gamma \times \gamma}}\downarrow &\swArrow^{\mu^{-1}}& \downarrow^{\mathrlap{\gamma}} \\ Ho(\mathcal{C}) \times Ho(\mathcal{C}) &\underset{\otimes^L}{\longrightarrow}& Ho(\mathcal{C}) } \,.

Then consider the associators:

The essential uniqueness of derived functors shows that the left derived functor of ()(()())(-)\otimes ( (-) \otimes (-) ) and of (()())()( (-) \otimes (-) )\otimes (-) is the composite of two applications of L\otimes^L, due to the factorization

𝒞 c×𝒞 c×𝒞 c ()(()()) 𝒞 c γ×γ×γ γ Ho(𝒞)×Ho(𝒞)×Ho(𝒞) 𝕃(()(()())) Ho(𝒞) \array{ \mathcal{C}_c \times \mathcal{C}_c \times \mathcal{C}_c &\overset{(-) \otimes ( (-) \otimes (-) )}{\longrightarrow}& \mathcal{C}_c \\ {}^{\mathllap{\gamma \times \gamma \times \gamma}}\downarrow &\swArrow& \downarrow^{\mathrlap{\gamma}} \\ Ho(\mathcal{C}) \times Ho(\mathcal{C}) \times Ho(\mathcal{C}) &\underset{\mathbb{L}((-) \otimes ( (-) \otimes (-) ))}{\longrightarrow}& Ho(\mathcal{C}) }
\;\;\;\;\;\;\; \simeq \;\;\;\;\;\;\;
𝒞 c×𝒞 c×𝒞 c id× 𝒞 c×𝒞 c 𝒞 c γ×γ×γ id×μ 1 γ×γ μ 1 γ Ho(𝒞)×Ho(𝒞)×Ho(𝒞) id× L Ho(𝒞)×Ho(𝒞) L Ho(𝒞) \array{ \mathcal{C}_c \times \mathcal{C}_c \times \mathcal{C}_c &\overset{id \times \otimes}{\longrightarrow}& \mathcal{C}_c \times \mathcal{C}_c &\overset{\otimes}{\longrightarrow}& \mathcal{C}_c \\ {}^{\mathllap{\gamma \times \gamma \times \gamma}}\downarrow &\swArrow_{id \times \mu^{-1}}& {}^{\mathllap{\gamma \times \gamma}}\downarrow &\swArrow_{\mu^{-1}}& \downarrow^{\mathrlap{\gamma}} \\ Ho(\mathcal{C}) \times Ho(\mathcal{C}) \times Ho(\mathcal{C}) &\underset{id \times \otimes^L}{\longrightarrow}& Ho(\mathcal{C}) \times Ho(\mathcal{C}) &\underset{\otimes^L}{\longrightarrow}& Ho(\mathcal{C}) }

and similarly for the case with the parenthesis to the left.

So let

𝒞 c×𝒞 c×𝒞 c (()())() 𝒞 γ×γ×γ μ 1(μ 1×id) γ Ho(𝒞)×Ho(𝒞)×Ho(𝒞) (() L()) L() Ho(𝒞),𝒞 c×𝒞 c×𝒞 c ()(()()) 𝒞 γ×γ×γ μ 1(id×μ 1) γ Ho(𝒞)×Ho(𝒞)×Ho(𝒞) () L(() L()) Ho(𝒞) \array{ \mathcal{C}_c \times \mathcal{C}_c \times \mathcal{C}_c &\overset{((-)\otimes(-))\otimes (-)}{\longrightarrow}& \mathcal{C} \\ {}^{ \mathllap{ \gamma \times \gamma \times \gamma } }\downarrow &\swArrow_{\mu^{-1}\cdot (\mu^{-1} \times id)}& \downarrow^{\mathrlap{\gamma}} \\ Ho(\mathcal{C}) \times Ho(\mathcal{C}) \times Ho(\mathcal{C}) &\overset{((-)\otimes^L(-))\otimes^L (-)}{\longrightarrow}& Ho(\mathcal{C}) } \;\;\;\,,\;\;\;\;\; \array{ \mathcal{C}_c \times \mathcal{C}_c \times \mathcal{C}_c &\overset{ (-) \otimes ( (-) \otimes (-) ) }{\longrightarrow}& \mathcal{C} \\ {}^{\mathllap{ \gamma \times \gamma \times \gamma } }\downarrow &\swArrow_{\mu^{-1}\cdot (id \times \mu^{-1})}& \downarrow^{\mathrlap{\gamma }} \\ Ho(\mathcal{C}) \times Ho(\mathcal{C}) \times Ho(\mathcal{C}) &\overset{ (-) \otimes^L ( (-) \otimes^L (-) ) }{\longrightarrow}& Ho(\mathcal{C}) }

be the [[natural isomorphism]] exhibiting the [[derived functors]] of the two possible tensor products of three objects, as shown at the top. By pasting the second with the [[associator]] natural isomorphism of 𝒞\mathcal{C} we obtain another such factorization for the first, as shown on the left below,

()𝒞 c×𝒞 c×𝒞 c (()())() 𝒞 = α = 𝒞 c×𝒞 c×𝒞 c ()(()()) 𝒞 γ×γ×γ μ 1(id×μ 1) γ Ho(𝒞)×Ho(𝒞)×Ho(𝒞) () L(() L()) Ho(𝒞)=𝒞 c×𝒞 c×𝒞 c (()())() 𝒞 γ×γ×γ μ 1(id×μ) γ Ho(𝒞)×Ho(𝒞)×Ho(𝒞) (() L()) L() Ho(𝒞) = α L = Ho(𝒞)×Ho(𝒞)×Ho(𝒞) () L(() L()) Ho(𝒞) (\star) \;\;\;\;\;\; \array{ \mathcal{C}_c \times \mathcal{C}_c \times \mathcal{C}_c &\overset{((-)\otimes(-))\otimes (-)}{\longrightarrow}& \mathcal{C} \\ {}^{\mathllap{=}}\downarrow &\swArrow_{\alpha}& \downarrow^{\mathrlap{=}} \\ \mathcal{C}_c \times \mathcal{C}_c \times \mathcal{C}_c &\overset{ (-) \otimes ( (-) \otimes (-) ) }{\longrightarrow}& \mathcal{C} \\ {}^{\mathllap{ \gamma \times \gamma \times \gamma }}\downarrow &\swArrow_{\mu^{-1} \cdot ( id \times \mu^{-1} )}& \downarrow^{\mathrlap{ \gamma }} \\ Ho(\mathcal{C}) \times Ho(\mathcal{C}) \times Ho(\mathcal{C}) &\overset{ (-) \otimes^L ( (-) \otimes^L (-) ) }{\longrightarrow}& Ho(\mathcal{C}) } \;\;\;\;\;\; = \;\;\;\;\;\; \array{ \mathcal{C}_c \times \mathcal{C}_c \times \mathcal{C}_c &\overset{((-)\otimes(-))\otimes (-)}{\longrightarrow}& \mathcal{C} \\ {}^{\mathllap{ \gamma \times \gamma \times \gamma }}\downarrow &\swArrow_{\mu^{-1}\cdot (id \times \mu)}& \downarrow^{\mathrlap{\gamma }} \\ Ho(\mathcal{C}) \times Ho(\mathcal{C}) \times Ho(\mathcal{C}) &\overset{((-)\otimes^L(-))\otimes^L (-)}{\longrightarrow}& Ho(\mathcal{C}) \\ {}^{\mathllap{=}}\downarrow &\swArrow_{\alpha^L}& \downarrow^{\mathrlap{=}} \\ Ho(\mathcal{C})\times Ho(\mathcal{C})\times Ho(\mathcal{C}) &\underset{(-)\otimes^L((-)\otimes^L (-))}{\longrightarrow}& Ho(\mathcal{C}) }

and hence by the universal property of the factorization through the derived functor, there exists a unique natural isomorphism α L\alpha^L such as to make this composite of natural isomorphisms equal to the one shown on the right. Hence the [[pentagon identity]] satisfied by α\alpha implies a pentagon identity for α L\alpha^L, and so α L\alpha^L is an [[associator]] for L\otimes^L.

Moreover, this equation of natural isomorphisms says that on components the following [[commuting diagram|diagram commutes]]

(γ(X) Lγ(Y)) Lγ(Z) α γ(X),γ(Y),γ(Z) L γ(X) L(γ(Y) Lγ(Z)) μ 1(μ 1×id) μ 1(id×μ 1) γ((XY)Z) γ(α) γ(X(YZ)). \array{ (\gamma(X) \otimes^L \gamma(Y)) \otimes^L \gamma(Z) &\overset{\alpha^L_{\gamma(X), \gamma(Y), \gamma(Z)}}{\longrightarrow}& \gamma(X) \otimes^L (\gamma(Y) \otimes^L \gamma(Z)) \\ {}^{\mathllap{\mu^{-1}\cdot (\mu^{-1} \times id)}}\uparrow && \uparrow^{\mu^{-1} \cdot (id\times \mu^{-1})} \\ \gamma( (X \otimes Y) \otimes Z ) &\underset{\gamma(\alpha)}{\longrightarrow}& \gamma(X \otimes (Y \otimes Z)) } \,.

This is just the [[coherence law]] for the the compatibility of the [[monoidal functor]] μ\mu with the associators.

Similarly consider now the [[unitors]].

The essential uniqueness of the derived functors gives that the left derived functor of 1()1\otimes (-) is γ(1) L()\gamma(1)\otimes^L (-)

𝒞 c 1() 𝒞 c γ γ Ho(𝒞) 𝕃(1()) Ho(𝒞)𝒞