symmetric monoidal (∞,1)-category of spectra
In suitable “coordinate-free” presentations of spectra, the structure of a (commutative) monoid with respect to the smash product of spectra (an A-infinity ring (E-infinity ring)) may be expressed directly as a lax monoidal functor on the indexing spaces, hence a functor that intertwines the smash product of indexing spaces with that of the component spaces, but without explicitly mentioning the smash product of spectra.
So a functor with smash products is a suitably well behaved functor
from a monoidal category $(\mathcal{D},\wedge)$ to pointed topological spaces/pointed simplicial sets and equipped with natural transformations
and
that are associative and unital in the evident sense.
For the case of highly structured spectra such as orthogonal spectra, symmetric spectra and S-modules, the equivalence of FSPs with monoids with respect to the symmetric smash product of spectra is due to this proposition at Day convolution. (MMSS 00, prop. 22.1, prop. 22.6).
(For instance accounts such as (Kochmann 96, section 3.3, Schwede 14) follow this perspective and define ring spectra first as FSPs, before or introducing the smash product on spectra)
For the monoidal model structure for excisive functors, the fact that monoids with respect to the symmetric smash product of spectra are equivalently FSPs is discussed in (Lydakis 98, remark 5.12). See this proposition.
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.
Let $\mathcal{C}, \mathcal{D}$ be pointed topologically enriched categories (def.), i.e. enriched categories over $(Top_{cg}^{\ast/}, \wedge, S^0)$ from example .
The pointed topologically enriched opposite category $\mathcal{C}^{op}$ is the topologically enriched category with the same objects as $\mathcal{C}$, with hom-spaces
and with composition given by braiding followed by the composition in $\mathcal{C}$:
the pointed topological product category $\mathcal{C} \times \mathcal{D}$ is the topologically enriched category whose objects are pairs of objects $(c,d)$ with $c \in \mathcal{C}$ and $d\in \mathcal{D}$, whose hom-spaces are the smash product of the separate hom-spaces
and whose composition operation is the braiding followed by the smash product of the separate composition operations:
A pointed topologically enriched functor (def.) into $Top^{\ast/}_{cg}$ (exmpl.) out of a pointed topological product category as in def.
(a “pointed topological bifunctor”) has component maps of the form
By functoriallity and under passing to adjuncts (cor.) this is equivalent to two commuting actions
and
In the special case of a functor out of the product category of some $\mathcal{C}$ with its opposite category (def. )
then this takes the form
and
Let $\mathcal{C}$ be a small pointed topologically enriched category (def.), i.e. an enriched category over $(Top_{cg}^{\ast/}, \wedge, S^0)$ from example . Let
be a pointed topologically enriched functor (def.) out of the pointed topological product category of $\mathcal{C}$ with its opposite category, according to def. .
The coend of $F$, denoted $\overset{c \in \mathcal{C}}{\int} F(c,c)$, is the coequalizer in $Top_{cg}^{\ast}$ (prop., exmpl., prop., cor.) of the two actions encoded in $F$ via example :
The end of $F$, denoted $\underset{c\in \mathcal{C}}{\int} F(c,c)$, is the equalizer in $Top_{cg}^{\ast/}$ (prop., exmpl., prop., cor.) of the adjuncts of the two actions encoded in $F$ via example :
Let $\mathcal{C}$ be a small pointed topologically enriched category (def.). For $F,G \;\colon\; \mathcal{C} \longrightarrow Top^{\ast/}_{cg}$ two pointed topologically enriched functors, then the end (def. ) of $Maps(F(-),G(-))_\ast$ is a topological space whose underlying pointed set is the pointed set of natural transformations $F\to G$ (def.)
The underlying pointed set functor $U\colon Top^{\ast/}_{cg}\to Set^{\ast/}$ preserves all limits (prop., prop., prop.). Therefore there is an equalizer diagram in $Set^{\ast/}$ of the form
Here the object in the middle is just the set of collections of component morphisms $\left\{ F(c)\overset{\eta_c}{\to} G(c)\right\}_{c\in \mathcal{C}}$. The two parallel maps in the equalizer diagram take such a collection to the functions which send any $c \overset{f}{\to} d$ to the result of precomposing
and of postcomposing
each component in such a collection, respectively. These two functions being equal, hence the collection $\{\eta_c\}_{c\in \mathcal{C}}$ being in the equalizer, means precisley that for all $c,d$ and all $f\colon c \to d$ the square
is a commuting square. This is precisley the condition that the collection $\{\eta_c\}_{c\in \mathcal{C}}$ be a natural transformation.
Conversely, example says that ends over bifunctors of the form $Maps(F(-),G(-)))_\ast$ constitute hom-spaces between pointed topologically enriched functors:
Let $\mathcal{C}$ be a small pointed topologically enriched categories (def.). Define the structure of a pointed topologically enriched category on the category $[\mathcal{C}, Top_{cg}^{\ast/}]$ of pointed topologically enriched functors to $Top^{\ast/}_{cg}$ (exmpl.) by taking the hom-spaces to be given by the ends (def. ) of example :
and by taking the composition maps to be the morphisms induced by the maps
by observing that these equalize the two actions in the definition of the end.
The resulting pointed topologically enriched category $[\mathcal{C},Top^{\ast/}_{cg}]$ is also called the $Top^{\ast/}_{cg}$-enriched functor category over $\mathcal{C}$ with coefficients in $Top^{\ast/}_{cg}$.
First of all this yields a concise statement of the pointed topologically enriched Yoneda lemma (prop.)
(topologically enriched Yoneda lemma)
Let $\mathcal{C}$ be a small pointed topologically enriched categories (def.). For $F \colon \mathcal{C}\to Top^{\ast/}_{cg}$ a pointed topologically enriched functor (def.) and for $c\in \mathcal{C}$ an object, there is a natural isomorphism
between the hom-space of the pointed topological functor category, according to def. , from the functor represented by $c$ to $F$, and the value of $F$ on $c$.
In terms of the ends (def. ) defining these hom-spaces, this means that
In this form the statement is also known as Yoneda reduction.
The proof of prop. is essentially dual to the proof of the next prop. .
Now that natural transformations are phrased in terms of ends (example ), as is the Yoneda lemma (prop. ), it is natural to consider the dual statement involving coends:
Let $\mathcal{C}$ be a small pointed topologically enriched categories (def.). For $F \colon \mathcal{C}\to Top^{\ast/}_{cg}$ a pointed topologically enriched functor (def.) and for $c\in \mathcal{C}$ an object, there is a natural isomorphism
Moreover, the morphism that hence exhibits $F(c)$ as the coequalizer of the two morphisms in def. is componentwise the canonical action
which is adjunct to the component map $\mathcal{C}(d,c) \to Maps(F(c),F(d))_{\ast}$ of the topologically enriched functor $F$.
(e.g. MMSS 00, lemma 1.6)
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
where two such pairs
are regarded as equivalent if there exists
such that
(Because then the two pairs are the two images of the pair $(g,x)$ under the two morphisms being coequalized.)
But now considering the case that $d = c_0$ and $g = id_{c_0}$, so that $f = \phi$ shows that any pair
is identified, in the coequalizer, with the pair
hence with $\phi(x)\in F(c_0)$.
This shows the claim at the level of the underlying sets. To conclude it is now sufficient (prop.) to show that the topology on $F(c_0) \in Top^{\ast/}_{cg}$ is the final topology (def.) of the system of component morphisms
which we just found. But that system includes
which is a retraction
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 $F(c)$.
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 $X$ a topological space, $f \colon X \to\mathbb{R}$ a continuous function and $\delta(-,x_0)$ denoting the Dirac distribution, then
It is this analogy that gives the name to the following statement:
(Fubini theorem for (co)-ends)
For $F$ a pointed topologically enriched bifunctor on a small pointed topological product category $\mathcal{C}_1 \times \mathcal{C}_2$ (def. ), i.e.
then its end and coend (def. ) is equivalently formed consecutively over each variable, in either order:
and
Because the pointed compactly generated mapping space functor (exmpl.)
takes colimits in the first argument and limits in the second argument to limits (cor.), it also takes coends in the first argument and ends in the second argument, to ends (def. ):
and
(left Kan extension via coends)
Let $\mathcal{C}, \mathcal{D}$ be small pointed topologically enriched categories (def.) and let
be a pointed topologically enriched functor (def.). Then precomposition with $p$ constitutes a functor
$G\mapsto G\circ p$. This functor has a left adjoint $Lan_p$, called left Kan extension along $p$
which is given objectwise by a coend (def. ):
Use the expression of natural transformations in terms of ends (example and def. ), then use the respect of $Maps(-,-)_\ast$ 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:
We recall the basic definitions of monoidal categories and of monoids and modules internal to monoidal categories. All examples are at the end of this section, starting with example below.
A (pointed) topologically enriched monoidal category is a (pointed) topologically enriched category $\mathcal{C}$ (def.) equipped with
a (pointed) topologically enriched functor (def.)
out of the (pointed) topologival product category of $\mathcal{C}$ with itself (def. ), called the tensor product,
an object
called the unit object or tensor unit,
called the associator,
called the left unitor, and a natural isomorphism
called the right unitor,
such that the following two kinds of diagrams commute, for all objects involved:
triangle identity:
the pentagon identity:
(Kelly 64)
Let $(\mathcal{C}, \otimes, 1)$ be a monoidal category, def. . Then the left and right unitors $\ell$ and $r$ satisfy the following conditions:
$\ell_1 = r_1 \;\colon\; 1 \otimes 1 \overset{\simeq}{\longrightarrow} 1$;
for all objects $x,y \in \mathcal{C}$ the following diagram commutes:
Analogously for the right unitor.
A (pointed) topological braided monoidal category, is a (pointed) topological monoidal category $\mathcal{C}$ (def. ) equipped with a natural isomorphism
called the braiding, such that the following two kinds of diagrams commute for all objects involved:
and
where $a_{x,y,z} \colon (x \otimes y) \otimes z \to x \otimes (y \otimes z)$ denotes the components of the associator of $\mathcal{C}^\otimes$.
A (pointed) topological symmetric monoidal category is a (pointed) topological braided monoidal category (def. ) for which the braiding
satisfies the condition:
for all objects $x, y$
Given a (pointed) topological symmetric monoidal category $\mathcal{C}$ with tensor product $\otimes$ (def. ) it is called a closed monoidal category if for each $Y \in \mathcal{C}$ the functor $Y \otimes(-)\simeq (-)\otimes X$ has a right adjoint, denoted $[Y,-]$
hence if there are natural isomorphisms
for all objects $X,Z \in \mathcal{C}$.
Since for the case that $X = 1$ is the tensor unit of $\mathcal{C}$ this means that
the object $[Y,Z] \in \mathcal{C}$ is an enhancement of the ordinary hom-set $Hom_{\mathcal{C}}(Y,Z)$ to an object in $\mathcal{C}$. Accordingly, it is also called the internal hom between $Y$ and $Z$.
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 $Top_{cg}$ 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.
The category $Top_{cg}^{\ast/}$ of pointed compactly generated topological spaces with tensor product the smash product $\wedge$ (def.)
is a symmetric monoidal category (def. ) with unit object the pointed 0-sphere $S^0$.
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 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 the archetypical case that motivates the notation “$\otimes$” for the pairing operation in a monoidal category:
A monoid in $(Ab, \otimes_{\mathbb{Z}}, \mathbb{Z})$ (def. ) is equivalently a ring.
A commutative monoid in in $(Ab, \otimes_{\mathbb{Z}}, \mathbb{Z})$ (def. ) is equivalently a commutative ring $R$.
An $R$-module object in $(Ab, \otimes_{\mathbb{Z}}, \mathbb{Z})$ (def. ) is equivalently an $R$-module;
The tensor product of $R$-module objects (def. ) is the standard tensor product of modules.
The category of module objects $R Mod(Ab)$ (def. ) is the standard category of modules $R Mod$.
Given a (pointed) topological monoidal category $(\mathcal{C}, \otimes, 1)$, then a monoid internal to $(\mathcal{C}, \otimes, 1)$ is
an object $A \in \mathcal{C}$;
a morphism $e \;\colon\; 1 \longrightarrow A$ (called the unit)
a morphism $\mu \;\colon\; A \otimes A \longrightarrow A$ (called the product);
such that
(associativity) the following diagram commutes
where $a$ is the associator isomorphism of $\mathcal{C}$;
(unitality) the following diagram commutes:
where $\ell$ and $r$ are the left and right unitor isomorphisms of $\mathcal{C}$.
Moreover, if $(\mathcal{C}, \otimes , 1)$ has the structure of a symmetric monoidal category (def. ) $(\mathcal{C}, \otimes, 1, B)$ with symmetric braiding $\tau$, then a monoid $(A,\mu, e)$ as above is called a commutative monoid in $(\mathcal{C}, \otimes, 1, B)$ if in addition
(commutativity) the following diagram commutes
A homomorphism of monoids $(A_1, \mu_1, e_1)\longrightarrow (A_2, \mu_2, f_2)$ is a morphism
in $\mathcal{C}$, such that the following two diagrams commute
and
Write $Mon(\mathcal{C}, \otimes,1)$ for the category of monoids in $\mathcal{C}$, and $CMon(\mathcal{C}, \otimes, 1)$ for its subcategory of commutative monoids.
Given a (pointed) topological monoidal category $(\mathcal{C}, \otimes, 1)$, then the tensor unit $1$ is a monoid in $\mathcal{C}$ (def. ) with product given by either the left or right unitor
By lemma , these two morphisms coincide and define an associative product with unit the identity $id \colon 1 \to 1$.
If $(\mathcal{C}, \otimes , 1)$ is a symmetric monoidal category (def. ), then this monoid is a commutative monoid.
Given a (pointed) topological monoidal category $(\mathcal{C}, \otimes, 1)$ (def. ), and given $(A,\mu,e)$ a monoid in $(\mathcal{C}, \otimes, 1)$ (def. ), then a left module object in $(\mathcal{C}, \otimes, 1)$ over $(A,\mu,e)$ is
an object $N \in \mathcal{C}$;
a morphism $\rho \;\colon\; A \otimes N \longrightarrow N$ (called the action);
such that
(unitality) the following diagram commutes:
where $\ell$ is the left unitor isomorphism of $\mathcal{C}$.
(action property) the following diagram commutes
A homomorphism of left $A$-module objects
is a morphism
in $\mathcal{C}$, such that the following diagram commutes:
For the resulting category of modules of left $A$-modules in $\mathcal{C}$ with $A$-module homomorphisms between them, we write
This is naturally a (pointed) topologically enriched category itself.
Given a monoidal category $(\mathcal{C},\otimes, 1)$ (def. ) with the tensor unit $1$ regarded as a monoid in a monoidal category via example , then the left unitor
makes every object $C \in \mathcal{C}$ into a left module, according to def. , over $C$. The action property holds due to lemma . This gives an equivalence of categories
of $\mathcal{C}$ with the category of modules over its tensor unit.
In the situation of def. , the monoid $(A,\mu, e)$ canonically becomes a left module over itself by setting $\rho \coloneqq \mu$. More generally, for $C \in \mathcal{C}$ any object, then $A \otimes C$ naturally becomes a left $A$-module by setting:
The $A$-modules of this form are called free modules.
The free functor $F$ constructing free $A$-modules is left adjoint to the forgetful functor $U$ which sends a module $(N,\rho)$ to the underlying object $U(N,\rho) \coloneqq N$.
A homomorphism out of a free $A$-module is a morphism in $\mathcal{C}$ of the form
fitting into the diagram (where we are notationally suppressing the associator)
Consider the composite
i.e. the restriction of $f$ to the unit “in” $A$. By definition, this fits into a commuting square of the form (where we are now notationally suppressing the associator and the unitor)
Pasting this square onto the top of the previous one yields
where now the left vertical composite is the identity, by the unit law in $A$. This shows that $f$ is uniquely determined by $\tilde f$ via the relation
This natural bijection between $f$ and $\tilde f$ establishes the adjunction.
Given a (pointed) topological symmetric monoidal category $(\mathcal{C}, \otimes, 1)$ (def. ), given $(A,\mu,e)$ a commutative monoid in $(\mathcal{C}, \otimes, 1)$ (def. ), and given $(N_1, \rho_1)$ and $(N_2, \rho_2)$ two left $A$-module objects (def.), then the tensor product of modules $N_1 \otimes_A N_2$ is, if it exists, the coequalizer
Given a (pointed) topological symmetric monoidal category $(\mathcal{C}, \otimes, 1)$ (def. ), and given $(A,\mu,e)$ a commutative monoid in $(\mathcal{C}, \otimes, 1)$ (def. ). If all coequalizers exist in $\mathcal{C}$, then the tensor product of modules $\otimes_A$ from def. makes the category of modules $A Mod(\mathcal{C})$ into a symmetric monoidal category, $(A Mod, \otimes_A, A)$ with tensor unit the object $A$ itself, regarded as an $A$-module via prop. .
Given a monoidal category of modules $(A Mod , \otimes_A , A)$ as in prop. , then a monoid $(E, \mu, e)$ in $(A Mod , \otimes_A , A)$ (def. ) is called an $A$-algebra.
Given a monoidal category of modules $(A Mod , \otimes_A , A)$ in a monoidal category $(\mathcal{C},\otimes, 1)$ as in prop. , and an $A$-algebra $(E,\mu,e)$ (def. ), then there is an equivalence of categories
between the category of commutative monoids in $A Mod$ and the coslice category of commutative monoids in $\mathcal{C}$ under $A$, hence between commutative $A$-algebras in $\mathcal{C}$ and commutative monoids $E$ in $\mathcal{C}$ that are equipped with a homomorphism of monoids $A \longrightarrow E$.
(e.g. EKMM 97, VII lemma 1.3)
In one direction, consider a $A$-algebra $E$ with unit $e_E \;\colon\; A \longrightarrow E$ and product $\mu_{E/A} \colon E \otimes_A E \longrightarrow E$. There is the underlying product $\mu_E$
By considering a diagram of such coequalizer diagrams with middle vertical morphism $e_E\circ e_A$, one find that this is a unit for $\mu_E$ and that $(E, \mu_E, e_E \circ e_A)$ is a commutative monoid in $(\mathcal{C}, \otimes, 1)$.
Then consider the two conditions on the unit $e_E \colon A \longrightarrow E$. First of all this is an $A$-module homomorphism, which means that
commutes. Moreover it satisfies the unit property
By forgetting the tensor product over $A$, the latter gives
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 pasted to the square $(\star)$ above, to yield a commuting square
This shows that the unit $e_A$ is a homomorphism of monoids $(A,\mu_A, e_A) \longrightarrow (E, \mu_E, e_E\circ e_A)$.
Now for the converse direction, assume that $(A,\mu_A, e_A)$ and $(E, \mu_E, e'_E)$ are two commutative monoids in $(\mathcal{C}, \otimes, 1)$ with $e_E \;\colon\; A \to E$ a monoid homomorphism. Then $E$ inherits a left $A$-module structure by
By commutativity and associativity it follows that $\mu_E$ coequalizes the two induced morphisms $E \otimes A \otimes E \underoverset{\longrightarrow}{\longrightarrow}{\phantom{AA}} E \otimes E$. Hence the universal property of the coequalizer gives a factorization through some $\mu_{E/A}\colon E \otimes_A E \longrightarrow E$. This shows that $(E, \mu_{E/A}, e_E)$ is a commutative $A$-algebra.
Finally one checks that these two constructions are inverses to each other, up to isomorphism.
Let $\mathcal{C}$ be a small pointed topological monoidal category (def. ) with tensor product denoted $\otimes_{\mathcal{C}} \;\colon\; \mathcal{C} \times\mathcal{C} \to \mathcal{C}$.
Then the Day convolution tensor product on the pointed topological enriched functor category $[\mathcal{C},Top^{\ast/}_{cg}]$ (def. ) is the functor
out of the pointed topological product category (def. ) given by the following coend (def. )
Let $Seq$ denote the category with objects the natural numbers, and only the zero morphisms and identity morphisms on these objects:
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 $X_\bullet \in [Seq, Top_{cg}^{\ast/}]$ is an $\mathbb{N}$-sequence of pointed topological spaces. Given two such, then their Day convolution according to def. is
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.
Let $\mathcal{C}$ be a small pointed topologically enriched category (def.). Its external tensor product is the pointed topologically enriched functor
given by
i.e.
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 $\otimes_{\mathcal{C}}$: there is a natural isomorphism
Hence the adjunction unit is a natural transformation of the form
This perspective is highlighted in (MMSS 00, p. 60).
By prop. we may compute the left Kan extension as the following coend:
The Day convolution $\otimes_{Day}$ (def. ) is universally characterized by the property that there are natural isomorphisms
Write
for the $Top^{\ast/}_{cg}$-Yoneda embedding, so that for $c\in \mathcal{C}$ any object, $y(c)$ is the corepresented functor $y(c)\colon d \mapsto \mathcal{C}(c,d)$.
For $\mathcal{C}$ a small pointed topological monoidal category (def. ), the Day convolution tensor product $\otimes_{Day}$ of def. makes the pointed topologically enriched functor category
a pointed topological monoidal category (def. ) with tensor unit $y(1)$ co-represented by the tensor unit $1$ of $\mathcal{C}$.
Regarding associativity, observe that
where we used the Fubini theorem for coends (prop. ) and then twice the co-Yoneda lemma (prop. ). An analogous formula follows for $X \otimes_{Day} (Y \otimes_{Day} Z)))(c)$, and so associativity follows via prop. from the associativity of the smash product and of the tensor product $\otimes_{\mathcal{C}}$.
To see that $y(1)$ is the tensor unit for $\otimes_{Day}$, use the Fubini theorem for coends (prop. ) and then twice the co-Yoneda lemma (prop. ) to get for any $X \in [\mathcal{C},Top^{\ast/}_{cg}]$ that
For $\mathcal{C}$ a small pointed topological monoidal category (def. ) with tensor product denoted $\otimes_{\mathcal{C}} \;\colon\; \mathcal{C} \times\mathcal{C} \to \mathcal{C}$, the monoidal category with Day convolution $([\mathcal{C},Top^{\ast/}_{cg}], \otimes_{Day}, y(1))$ from def. is a closed monoidal category (def. ). Its internal hom $[-,-]_{Day}$ is given by the end (def. )
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:
In the situation of def. , the Yoneda embedding $c\mapsto \mathcal{C}(c,-)$ constitutes a strong monoidal functor
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
Let $(\mathcal{C},\otimes_{\mathcal{C}}, 1_{\mathcal{C}})$ and $(\mathcal{D},\otimes_{\mathcal{D}}, 1_{\mathcal{D}} )$ be two (pointed) topologically enriched monoidal categories (def. ). A topologically enriched lax monoidal functor between them is
a topologically enriched functor
a morphism
for all $x,y \in \mathcal{C}$
satisfying the following conditions:
(associativity) For all objects $x,y,z \in \mathcal{C}$ the following diagram commutes
where $a^{\mathcal{C}}$ and $a^{\mathcal{D}}$ denote the associators of the monoidal categories;
(unitality) For all $x \in \mathcal{C}$ the following diagrams commutes
and
where $\ell^{\mathcal{C}}$, $\ell^{\mathcal{D}}$, $r^{\mathcal{C}}$, $r^{\mathcal{D}}$ denote the left and right unitors of the two monoidal categories, respectively.
If $\epsilon$ and alll $\mu_{x,y}$ are isomorphisms, then $F$ is called a strong monoidal functor.
If moreover $(\mathcal{C},\otimes_{\mathcal{C}}, 1_{\mathcal{C}})$ and $(\mathcal{D},\otimes_{\mathcal{D}}, 1_{\mathcal{D}} )$ are equipped with the structure of braided monoidal categories (def. ), then the lax monoidal functor $F$ is called a braided monoidal functor if in addition the following diagram commutes for all objects $x,y \in \mathcal{C}$
A homomorphism $f\;\colon\; (F_1,\mu_1, \epsilon_1) \longrightarrow (F_2, \mu_2, \epsilon_2)$ between two (braided) lax monoidal functors is a monoidal natural transformation, in that it is
compatible with the product and the unit in that the following diagrams commute for all objects $x,y \in \mathcal{C}$:
and
We write $MonFun(\mathcal{C},\mathcal{D})$ for the resulting category of lax monoidal functors between monoidal categories $\mathcal{C}$ and $\mathcal{D}$, similarly $BraidMonFun(\mathcal{C},\mathcal{D})$ for the category of braided monoidal functors between braided monoidal categories, and $SymMonFun(\mathcal{C},\mathcal{D})$ for the category of braided monoidal functors between symmetric monoidal categories.
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 $(\mathcal{C},\otimes_{\mathcal{C}}, 1_{\mathcal{C}})$ and $(\mathcal{D},\otimes_{\mathcal{D}}, 1_{\mathcal{D}} )$ are symmetric monoidal categories (def. ) then a braided monoidal functor (def. ) between them is often called a symmetric monoidal functor.
Let $(\mathcal{C},\otimes_{\mathcal{C}}, 1_{\mathcal{C}})$ and $(\mathcal{D},\otimes_{\mathcal{D}}, 1_{\mathcal{D}} )$ be two (pointed) topologically enriched monoidal categories (def. ), and let $F \;\colon\; \mathcal{C} \longrightarrow \mathcal{D}$ be a topologically enriched lax monoidal functor between them, with product operation $\mu$.
Then a left module over the lax monoidal functor is
a topologically enriched functor
such that
(action property) For all objects $x,y,z \in \mathcal{C}$ the following diagram commutes
A homomorphism $f\;\colon\; (G_1, \rho_1) \longrightarrow (G_2,\rho_2)$ between two modules over a monoidal functor $(F,\mu,\epsilon)$ is
compatible with the action in that the following diagram commutes for all objects $x,y \in \mathcal{C}$:
We write $F Mod$ for the resulting category of modules over the monoidal functor $F$.
Let $(\mathcal{C},\otimes I)$ be a pointed topologically enriched category (symmetric monoidal category) monoidal category (def. ). Regard $(Top_{cg}^{\ast/}, \wedge , S^0)$ as a topological symmetric monoidal category as in example .
Then (commutative) monoids in (def. ) the Day convolution monoidal category $([\mathcal{C}, Top^{\ast/}_{cg}], \otimes_{Day}, y(1_{\mathcal{C}}))$ of prop. are equivalent to (braided) lax monoidal functors (def. ) of the form
called functors with smash products on $\mathcal{C}$, i.e. there are equivalences of categories of the form
Moreover, module objects over these monoid objects are equivalent to the corresponding modules over monoidal functors.
This is stated in some form in (Day 70, example 3.2.2). It is highlighted again in (MMSS 00, prop. 22.1).
By definition , a lax monoidal functor $F \colon \mathcal{C} \to Top^{\ast/}_{cg}$ is a topologically enriched functor equipped with a morphism of pointed topological spaces of the form
and equipped with a natural system of maps of pointed topological spaces of the form
for all $c_1,c_2 \in \mathcal{C}$.
Under the Yoneda lemma (prop. ) the first of these is equivalently a morphism in $[\mathcal{C}, Top^{\ast/}_{cg}]$ of the form
Moreover, under the natural isomorphism of corollary the second of these is equivalently a morphism in $[\mathcal{C}, Top^{\ast/}_{cg}]$ of the form
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 object $F$ under $\otimes_{Day}$.
Similarly for module objects and modules over monoidal functors.
Let $f \;\colon\; \mathcal{C} \longrightarrow \mathcal{D}$ be a lax monoidal functor (def. ) between pointed topologically enriched monoidal categories (def. ). Then the induced functor
given by $(f^\ast X)(c)\coloneqq X(f(c))$ preserves monoids under Day convolution
Moreover, if $\mathcal{C}$ and $\mathcal{D}$ are symmetric monoidal categories (def. ) and $f$ is a braided monoidal functor (def. ), then $f^\ast$ also preserves commutative monoids
This is an immediate corollary of prop. , since the composite of two (braided) lax monoidal functors is itself canonically a (braided) lax monoidal functor.
Write
for the full subcategory of 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. )
is the category of pre-excisive functors.
Write
for the functor co-represented by 0-sphere. This is equivalently the inclusion $\iota_{fin}$ itself:
We call this the standard incarnation of the sphere spectrum as a pre-excisive functor.
By prop. the smash product of pointed compactly generated topological spaces induces the structure of a closed (def. ) symmetric monoidal category (def. )
with
tensor unit the sphere spectrum $\mathbb{S}_{exc}$;
tensor product the Day convolution product $\otimes_{Day}$ from def. ,
called the symmetric monoidal smash product of spectra for the model of pre-excisive functors;
internal hom the dual operation $[-,-]_{Day}$ from prop. ,
called the mapping spectrum construction for pre-excisive functors.
By example the sphere spectrum incarnated as a pre-excisive functor $\mathbb{S}_{exc}$ (according to def. ) is canonically a commutative monoid in the category of pre-excisive functors (def. )
Moreover, by example , every object of $Exc(Top_{cg})$ (def. ) is canonically a module object over $\mathbb{S}_{exc}$. We may therefore tautologically identify the category of pre-excisive functors with the module category over the sphere spectrum:
We now consider restricting the domain of the pre-excisive functors of def. .
Define the following pointed topologically enriched (def.) symmetric monoidal categories (def. ):
$Seq$ 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
The tensor product is the addition of natural numbers, $\otimes = +$, and the tensor unit is 0.
$Sym$ is the standard skeleton of the core of FinSet with zero morphisms adjoined: its objects are the finite sets $\{1, \cdots,n\}$ for $n \in \mathbb{N}$, all non-zero morphisms are automorphisms and the automorphism group of $\{1,\cdots,n\}$ is the symmetric group $\Sigma_n$, hence the hom-spaces are the following discrete topological spaces:
The tensor product is the disjoint union of sets, tensor unit is the empty set.
$Orth$ has as objects finite dimenional real linear inner product spaces $(V, \langle -,-\rangle)$ and as non-zero morphisms the linear isometric isomorphisms between these; hence the automorphism group of the object $(V, \langle -,-\rangle)$ is the orthogonal group $O(V)$; the monoidal product is direct sum of linear spaces, the tensor unit is the 0-vector space; again we turn this into a $Top^{\ast/}$-enriched category by adjoining a basepoint to the hom-spaces;
There is a sequence of canonical faithful pointed topological subcategory inclusions
into the pointed topological category of pointed compactly generated topological spaces of finite CW-type (def. ).
Here $S^V$ denotes the one-point compactification of $V$. On morphisms $sym \colon (\Sigma_n)_+ \hookrightarrow (O(n))_+$ is the canonical inclusion of permutation matrices into orthogonal matrices and $orth \colon O(V)_+ \hookrightarrow Aut(S^V)$ is on $O(V)$ the topological subspace inclusions of the pointed homeomorphisms $S^V \to S^V$ that are induced under forming one-point compactification from linear isometries of $V$ (“representation spheres”).
Consider the sequence of restrictions of topological diagram categories, according to prop. along the above inclusions:
Write
for the restriction of the excisive functor incarnation of the sphere spectrum (from def. ) along these inclusions.
Since $\mathbb{S}_{exc}$ 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 prop. , all these restricted sphere spectra are still monoids. However, while $orth$ and $sym$ are braided monoidal functors, the functor $seq$ is not braided, hence $\mathbb{S}_{orth}$ and $\mathbb{S}_{sym}$ are commutative monoids, but $\mathbb{S}_{Seq}$ is not commutative.
$\mathbb{S}$ | $\mathbb{S}_{Orth}$ | $\mathbb{S}_{Sym}$ | $\mathbb{S}_{Seq}$ | |
---|---|---|---|---|
monoid | yes | yes | yes | yes |
commutative monoid | yes | yes | yes | no |
tensor unit | yes | no | no | no |
Therefore we may consider module objects over the restrictions of the sphere spectrum from def. .
The categories of modules (def. ) over $\mathbb{S}_{Orth}$, $\mathbb{S}_{Sym}$ and $\mathbb{S}_{Seq}$ (def. ) are equivalent, respectively, to the categories of orthogonal spectra, symmetric spectra and sequential spectra (in compactly generated topological spaces):
Write $\mathbb{S}_{dia}$ for any of the three monoids. By prop. , left modules with respect to Day convolution are equivalently modules over monoidal functors over the monoidal functor corresponding to $\mathbb{S}_{dia}$. This means that for $\mathbb{S}_{Sym}$ and $\mathbb{S}_{Seq}$ they are functors $X \colon Sym \longrightarrow sSet^{\ast/}$ or $X \colon Seq \longrightarrow sSet^{\ast/}$, respectively equipped with natural transformations
satisfying the evident categorified action property. In the present case this action property says that these morphisms are determined by
under the isomorphisms $S^p \simeq S^1 \wedge S^{p-1}$. Naturality of all these morphisms as functors on $Sym$ is the equivariance under the symmetric group actions in the definition of symmetric spectra.
Similarly, modules over $\mathbb{S}_{Orth}$ are equivalently functors
etc. and their functoriality embodies the orthogonal group-equivariance in the definition of orthogonal spectra.
Consider the non-full inclusion of topologically enriched categories
on the standard n-spheres $S^n \coloneqq (S^1)^{\wedge^n}$, with hom-spaces given by the orthogonal groups with basepoint adjoint, acting on these spheres as their canonical representation spheres
Regard Orth as a monoidal category with monoidal structure induced form $(Top^{\ast/}_{cg}, \wedge, S^0)$ (via example ) under the restriction. This makes the inclusion a braided monoidal functor.
Restricting the standard pre-excisive model $y(S^0)$ of the sphere spectrum yields $\mathbb{S}_{orth}$. Since restriction is a monoidal functor, and since $y(S^0)$ is the tensor unit and hence canonically a monoid, prop. says that $\mathbb{S}_{orth}$ is still a commutative monoid with respect to Day convolution:
The category of orthogonal spectra is the category of $\mathbb{S}_{orth}$-modules (def. ):
Since $\mathbb{S}_{orth}$ is a commutative monoid, prop. says that there is a symmetric monoidal category structure $\otimes_{\mathbb{S}_{orth}}$ on $OrthSpec(Top_{cg})$. This is the symmetric monoidal smash product of spectra for orthogonal spectra.
An orthogonal ring spectrum $E$ is a monoid with respect to $\otimes_{\mathbb{S}_{orth}}$, hence an $\mathbb{S}_{orth}$-algebra (def. ). By prop. , such $E$ is equivalently a monoid with respect to $\otimes_{Day}$ and equipped with a monoid homomorphism $\mathbb{S}_{orth} \longrightarrow E$. Finally, by prop. this is equivalently a functor with smash products
equipped with a natural transformation of functors with smash product
In the terminology of MMSS 00, def. 22.5 this is an “$Orth$-FSP over $\mathbb{S}_{Orth}$”.
Restrict further along the non-full inclusion
where $Sym$ has the same objects, but the hom-spaces are now just the symmetric groups (with basepoint adjoint)
Then proceed as for orthogonal spectra.
Restrict further along
where $Seq$ still has the same objects, the $n$-spheres, but no non-trivial morphisms (just the identity morphisms and the zero morphisms).
Now the inclusion $Seq \longrightarrow Top^{\ast/}_{cg}$ is no longer a braided monoidal functor, for the braiding on $Seq$ is trivial, while on $Top^{\ast/}_{cg}$ it is not. Accordingly the assumption of the second clause in prop. is vialoted.
Indeed, restricting $\mathbb{S}$ along this inclusion yields the stndard sequential sphere spectrum $\mathbb{S}_{seq}$ which is still a monoid with respect to Day convolution, but not a commutative monoid anymore (see at smash product of spectra – graded commutativity) and hence the assumption of prop. is violated.
The $\mathbb{S}_{seq}$-module objects (def. ) are equivalently the sequential spectra.
But since $\mathbb{S}_{seq}$ is not a commutative monoid, the assumption of prop. there is no induced tensor product on $\mathbb{S}_{seq}Mod$ and hence the story ends here.
The concept was introduced (before symmetric smash products of spectra had been found) in
Restricted to spheres as “FSPs defined on spheres” they were considered in
and identified there as the monoids in symmetric spectra as previously introduced by Jeff Smith.
In the model structure for excisive functors the concept was recovered in
and in the model of connective spectra by Gamma-spaces in
A systematic account is in
Based on discussion in
Last revised on August 2, 2024 at 17:00:31. See the history of this page for a list of all contributions to it.