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Stable Homotopy theory

This page collects material related to

on a unified construction and comparison of the model structure on sequential spectra, model structure on symmetric spectra, model structure on orthogonal spectra, model structure on excisive functors.

Related references include


Contents

Part I. Diagram spaces and diagram spectra

This part gives 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 sub-sites of the canonical representative of the sphere spectrum as an excisive functor on Top fin */Top^{\ast/}_{fin}.

Throughout, write

Throughout part I we are dealing with (Top */,)(Top^{\ast/},\wedge)-enriched categories, (Top */,)(Top^{\ast/}, \wedge)-enriched functors, etc., and then in part II we are dealing with (Top Quillen */,)(Top_{Quillen}^{\ast/}, \wedge)-enriched model categories etc.

𝕊\mathbb{S}-modules

Definition

Define the following pointed topologically enriched symmetric closed monoidal categories (the tensor product is a pointed topologically enriched functor):

  1. SeqSeq has as objects the natural numbers and has only identity morphisms, tensor product is the addition of natural numbers, tensor unit is 0. As a Top */Top^{\ast/}-enriched category the hom-spaces are

    Seq(n 1,n 2)={S 0 forn 1=n 2 * otherwise Seq(n_1,n_2) = \left\{ \array{ S^0 & for\; n_1 = n_2 \\ \ast & otherwise } \right.
  2. SymSym is the standard skeleton of the core of FinSet, objects are the sets {1,,n}\{1, \cdots,n\} for nn \in \mathbb{N}, all morphisms are automorphisms and the automorphism group of {1,,n}\{1,\cdots,n\} is the symmetric group Σ n\Sigma_n, tensor product is the disjoint union of sets, tensor unit is the empty set; we turn this into a Top */Top^{\ast/}-enriched category by adjoining a basepoint:

    Sym(n 1,n 2)={(Σ n 1) + forn 1=n 2 * otherwise Sym(n_1, n_2) = \left\{ \array{ (\Sigma_{n_1})_+ & for \; n_1 = n_2 \\ \ast & otherwise } \right.
  3. OrthOrth has as objects finite dimenional real linear inner product spaces (V,,)(V, \langle -,-\rangle) and as morphisms the linear isometric isomorphisms between these; hence the automorphism group of the object (V,,)(V, \langle -,-\rangle) is the orthogonal group O(V)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 */Top^{\ast/}-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) \simeq \left\{ \array{ O(V_1)_+ & for \; dim(V_1) = dim(V_2) \\ \ast & otherwise } \right.
  1. Top fin */Top_{fin}^{\ast/} is the full subcategory of pointed topological space on those homeomorphic to finite CW-complexes, tensor product is their smash product, tensor unit is the 0-sphere S 0S^0.

Denote the canonical faithful subcategory inclusions by

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

where 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 inclusion of permutation matrices into 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.

Proposition

The sequence of inclusions in def. 1 satisfies the following properties:

  1. All three inclusions are strong monoidal functors.

  2. Under passing to enriched functor categories, restriction () *(-)^\ast along these inclusions and left Kan extension () !(-)_! along them yields a sequence of adjunctions

    [Top fin */,Top */]orth *orth ![Orth,Top */]sym *sym ![Sym,Top */]seq *seq ![Seq,Top */]. \array{ [Top_{fin}^{\ast/}, Top^{\ast/}] \stackrel{\overset{orth_!}{\longleftarrow}}{\underset{orth^\ast}{\longrightarrow}} [Orth, Top^{\ast/}] \stackrel{\overset{sym_!}{\longleftarrow}}{\underset{sym^\ast}{\longrightarrow}} [Sym, Top^{\ast/}] \stackrel{\overset{seq_!}{\longleftarrow}}{\underset{seq^\ast}{\longrightarrow}} [Seq, Top^{\ast/}] } \,.
  3. All four enriched functor categories become symmetric monoidal categories with the Day convolution monoidal product structure induced by the monoidal structure of their sites.

  4. With respect to this all adjunctions above are symmetric monoidal adjunctions (the right adjoint is a symmetric lax monoidal functor, the left adjoint is even a symmetric strong monoidal functor).

(e.g. MMSS 00, I.3)

Notice the following:

Lemma

For 𝒞\mathcal{C} a VV-enriched monoidal category, under the Yoneda embedding

y:𝒞 op[𝒞,Top */] y \colon \mathcal{C}^{op} \hookrightarrow [\mathcal{C}, Top^{\ast/}]

the tensor unit in 𝒞\mathcal{C} goes to the tensor unit of the induced Day convolution structure on [𝒞,Top */][\mathcal{C}, Top^{\ast/}].

(see at Day convolution this lemma)

Definition

Write

𝕊y(S 0)[Top fin */,Top */] \mathbb{S} \coloneqq y(S^0) \in [Top_{fin}^{\ast/}, Top^{\ast/}]

for the image under the Yoneda embedding of the tensor unit in Top fin */Top_{fin}^{\ast/} with its smash product (the 0-sphere), which by lemma 1 is the tensor unit in ([Top fin */,Top */], Day)([Top_{fin}^{\ast/}, Top^{\ast/}], \otimes_{Day}).

Since this is going to be the standard presentation of the sphere spectrum in the model structure for excisive functors on [Top fin */,Top */][Top_{fin}^{\ast/}, Top^{\ast/}] we refer to it as the sphere spectrum.

For its restrictions along the above sub-site inclusions, prop. 1, write

𝕊 Orthorth *𝕊,𝕊 Symsym *𝕊 orth,𝕊 Seqseq *𝕊 sym. \mathbb{S}_{Orth} \coloneqq orth^\ast \mathbb{S} \,, \; \mathbb{S}_{Sym} \coloneqq sym^\ast \mathbb{S}_{orth} \,, \; \mathbb{S}_{Seq} \coloneqq seq^\ast \mathbb{S}_{sym} \,.
Remark

While 𝕊\mathbb{S} in def. 2 is the tensor unit in ([Top fin */,Top */], Day)([Top_{fin}^{\ast/}, Top^{\ast/}], \otimes_{Day}), neither of its restrictions 𝕊 Orth,𝕊 Sym,𝕊 Seq\mathbb{S}_{Orth},\mathbb{S}_{Sym}, \mathbb{S}_{Seq} is the tensor unit in ([Orth,Top */], Day),([Sym,Top */], Day),([Seq,Top */], Day)([Orth, Top^{\ast/}],\otimes_{Day}), ([Sym, Top^{\ast/}],\otimes_{Day}), ([Seq, Top^{\ast/}],\otimes_{Day}), respectively.

Nevertheless, because by prop. 1 the restriction functors are strong monoidal functors and because the tensor unit 𝕊\mathbb{S} canonically has the structure of a monoid object, each of 𝕊 Orth,𝕊 Sym,𝕊 Seq\mathbb{S}_{Orth},\mathbb{S}_{Sym}, \mathbb{S}_{Seq} inherts the structure of a monoid object in the respective Day convolution monoidal category.

Moreover, 𝕊 Orth\mathbb{S}_{Orth} and 𝕊 Sym\mathbb{S}_{Sym} are commutative monoid objects, while 𝕊 Seq\mathbb{S}_{Seq} is not commutative (due to the graded commutativity in the smash product of spheres which is not reflected in the trivial symmetry of the tensor product on SeqSeq).

𝕊\mathbb{S}𝕊 Orth\mathbb{S}_{Orth}𝕊 Sym\mathbb{S}_{Sym}𝕊 Seq\mathbb{S}_{Seq}
monoid objectyesyesyesyes
commutative monoid objectyesyesyesno
tensor unityesnonono

Explicitly, by the discussion at Day convolutions – Properties – Monoids, monoids with respect to Day convolution are equivalently lax monoidal functors on the site, and as such 𝕊 Orth\mathbb{S}_{Orth} is the one given by the canonical natural transformations

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

and 𝕊 Sym\mathbb{S}_{Sym} and 𝕊 Seq\mathbb{S}_{Seq} the ones given by the canonical natural transformations

S n 1S n 2S n 1+n 2. S^{n_1} \wedge S^{n_2} \longrightarrow S^{n_1 + n_2} \,.

Therefore we may consider module objects over the restrictions of the sphere spectrum from def. 2.

Proposition

The category of right module objects over 𝕊 Orth\mathbb{S}_{Orth}, 𝕊 Sym\mathbb{S}_{Sym} and 𝕊 Seq\mathbb{S}_{Seq} from def. 2, which are monoid objects by prop. 1, remark 1, are equivalent, respectively, to the categories of orthogonal spectra, symmetric spectra and sequential spectra (in compactly generated topological spaces):

𝕊 OrthMod rOrthSpec(Top) \mathbb{S}_{Orth} Mod_r \simeq OrthSpec(Top)
𝕊 SymMod rSymSpec(Top) \mathbb{S}_{Sym} Mod_r \simeq SymSpec(Top)
𝕊 SeqMod rSeqSpec(Top). \mathbb{S}_{Seq} Mod_r \simeq SeqSpec(Top) \,.
Proof

Write 𝕊 dia\mathbb{S}_{dia} for any of the three monoids. By the discussion at Day convolutions – Properties – Monoids, right modules with respect to Day convolution are equivalently right modules over monoidal functors over the monoidal functor corresponding to 𝕊 dia\mathbb{S}_{dia} as in remark 1. This means that for 𝕊 Sym\mathbb{S}_{Sym} and 𝕊 Seq\mathbb{S}_{Seq} they are functors X:SymsSet */X \colon Sym \longrightarrow sSet^{\ast/} or X:SeqsSet */X \colon Seq \longrightarrow sSet^{\ast/}, respectively equipped with natural transformations

X pS qX p+q X_p \wedge S^{q} \longrightarrow X_{p+q}

satisfying the evident categorified action property. In the present case this action property says that these morphisms are determined by

X pS 1X p+1 X_p \wedge S^1 \longrightarrow X_{p+1}

under the isomorphisms S pS 1S p1S^p \simeq S^1 \wedge S^{p-1}. Naturality of all these morphisms as functors on SymSym is the equivariance under the symmetric group actions in the definition of symmetric spectra.

Similarly, modules over 𝕊 Orth\mathbb{S}_{Orth} are equivalently functors

X VS WX VW X_V \wedge S^{W} \longrightarrow X_{V \oplus W}

etc. and their functoriality embodies the orthogonal group-equivariance in the definition of orthogonal spectra.

Remark

For completeness, we may, trivially, add to the three statements in prop. 2 the equivalence

𝕊Mod r[Top fin */,Top */], \mathbb{S} Mod_r \simeq [Top^{\ast/}_{fin}, Top^{\ast/}] \,,

which holds tautologically because by def. 2 𝕊\mathbb{S} is in fact the tensor unit in ([Top fin */,Top */], Day)([Top^{\ast/}_{fin}, Top^{\ast/}],\otimes_{Day}), so that every object here is canonically a module object over 𝕊\mathbb{S}.

Now the model structure on excisive functors shows that the category [Top fin */,Top */][Top^{\ast/}_{fin}, Top^{\ast/}] constitutes a model for stable homotopy theory, while this is not the case for either of its restrictions.

Hence we may read prop. 2 as saying that while restricting the domain of excisive functors breaks their property of being a model for stable homotopy theory, but at the same time retaining the correspondingly restricted sphere spectrum-module structure first of all becomes non-tautological after restriction and second restores the property of the objects to model spectra.

Definition

By remark 1 the categories 𝕊 SymMod r\mathbb{S}_{Sym} Mod_r, 𝕊 OrthMod r\mathbb{S}_{Orth} Mod_r and 𝕊 OrthMod r\mathbb{S}_{Orth} Mod_r are categories of modules over a commutative monoid object and as such they inherit symmetric monoidal category structure

(𝕊 diaMod, 𝕊 dia) (\mathbb{S}_{dia} Mod, \wedge_{\mathbb{S}_{dia}})

Via prop. 2 this is equivalently symmetric monoidal product structure

(SymSpec(Top),) (SymSpec(Top), \wedge)

on the category of symmetric spectra and

(OrthSpec(Top),) (OrthSpec(Top), \wedge)

on that of orthogonal spectra. This is called the symmetric monoidal smash product of spectra.

Remark

Combined with the free-forgetful adjunctions for module objects (free modules \dashv underlying objects) the situation described by prop. 1 and prop. 2 jointly is the following diagram of adjunctions

OrthSpec(Top) SymSpec(Top) SeqSpec(Top) 𝕊Mod r 𝕊 OrthMod r 𝕊 SymMod r 𝕊 SeqMod r U F U F U F U F [Top fin */,Top */] orth *orth ! [Orth,Top */] sym *sym ! [Sym,Top */] seq *seq ! [Seq,Top */]. \array{ && OrthSpec(Top) && SymSpec(Top) && SeqSpec(Top) \\ && \downarrow^{\mathrlap{\simeq}} && \downarrow^{\mathrlap{\simeq}} && \downarrow^{\mathrlap{\simeq}} \\ \mathbb{S} Mod_r && \mathbb{S}_{Orth} Mod_r && \mathbb{S}_{Sym} Mod_r && \mathbb{S}_{Seq} Mod_r \\ {}^{\mathllap{U}}\downarrow \uparrow^{\mathrlap{F}} && {}^{\mathllap{U}}\downarrow \uparrow^{\mathrlap{F}} && {}^{\mathllap{U}}\downarrow \uparrow^{\mathrlap{F}} && {}^{\mathllap{U}}\downarrow \uparrow^{\mathrlap{F}} \\ [Top_{fin}^{\ast/}, Top^{\ast/}] &\stackrel{\overset{orth_!}{\longleftarrow}}{\underset{orth^\ast}{\longrightarrow}}& [Orth, Top^{\ast/}] &\stackrel{\overset{sym_!}{\longleftarrow}}{\underset{sym^\ast}{\longrightarrow}}& [Sym, Top^{\ast/}] &\stackrel{\overset{seq_!}{\longleftarrow}}{\underset{seq^\ast}{\longrightarrow}}& [Seq, Top^{\ast/}] } \,.

In order to conveniently speak about all columns of the system of adjunctions in remark 3 in a unified way, we introduce the following notation.

Definition

We write dia{Top */,Orth,Sym,Seq}dia \in \{Top^{\ast/, Orth, Sym, Seq}\} generically for any one of the four sites in def. 1.

Accordingly we write 𝕊 dia{𝕊,𝕊 Orth,𝕊 Sym,𝕊 Seq}\mathbb{S}_{dia} \in \{\mathbb{S}, \mathbb{S}_{Orth}, \mathbb{S}_{Sym}, \mathbb{S}_{Seq}\} generically for any one of the four incarnations of the sphere spectrum according to def. 2, over these sites.

Finally we will write

𝕊 diaModseq *seq!𝕊 SeqModSeqSpec(Top) \mathbb{S}_{dia}Mod \stackrel{\overset{seq!}{\longleftarrow}}{\underset{seq^\ast}{\longrightarrow}} \mathbb{S}_{Seq}Mod \simeq SeqSpec(Top)

for the composition of the sequence of adjunctions to the right of the corresponding category of modules in the diagram below in prop. 3, regarded via prop. 2 and landing in the category of sequential spectra.

We would like to lift the horizontal adjunctions in the diagram in remark 3 from the bottom row to the top row. To that end, oberve that the categories of modules involved here are themselves still enriched functor categories:

Lemma

Let dia{Top */,Orth,Sym,Seq}{dia} \in \{Top^{\ast/}, {Orth}, {Sym}, {Seq}\} be any one of the sites in def. 1.

  1. There is an equivalence of categories

    𝕊 diaMod[𝕊 diaFreeMod op,Top */] \mathbb{S}_{dia} Mod \simeq [ \mathbb{S}_{dia} FreeMod^{op}, Top^{\ast/} ]

    between the corresponding category of modules (prop. 2) and the Top */Top^{\ast/}-enriched functor category over the opposite of its full subcategory on the free modules.

  2. If 𝕊 dia\mathbb{S}_{dia} is a commutative monoid (hence for dia{Top */,Orth,Sym}dia \in \{Top^\ast/, Orth,Sym\} but not for dia=Seqdia = Seq) then 𝕊 diaFreeMod op\mathbb{S}_{dia} FreeMod^{op} carries a symmetric monoidal category structure such that under the above equivalence its Day convolution is the tensor product of modules.

  3. There is a Top */Top^{\ast/}-enriched functor

    δ:dia𝕊 diaFreeMod op \delta \colon dia \longrightarrow \mathbb{S}_{dia} FreeMod^{op}

    which is monoidal when 𝕊 dia\mathbb{S}_{dia} is commutative and which is such that the (free module\dashvforgetful functor)-adjunction is equivalently the base change along δ\delta:

    𝕊 diaMod [𝕊 diaFreeMod op,Top */] U F δ * δ ! [dia,Top */] = [dia,Top */]. \array{ \mathbb{S}_{dia} Mod &\simeq& [ \mathbb{S}_{dia} FreeMod^{op}, Top^{\ast/}] \\ {}^{\mathllap{U}}\downarrow \uparrow^{\mathrlap{F}} && {}^{\mathllap{\delta^\ast}}\downarrow \uparrow^{\mathrlap{\delta_!}} \\ [dia, Top^{\ast/}] &=& [dia, Top^{\ast/}] } \,.

This is a general statement about modules with respect to Day convolution monoidal structures, see this proposition (MMSS 00, theorem 2.2).

Example

For the sequential case dia=Seqdia = Seq, then the opposite category 𝕊 SeqFreeMod op\mathbb{S}_{Seq} FreeMod^{op} of free modules over 𝕊 Seq\mathbb{S}_{Seq} (def. 2) in lemma 2 is identified as the non-full subcategory StdSpheresStdSpheres of Top */Top^{\ast/} whose objects are the standard spheres S n(S 1) nS^n \coloneqq (S^1)^{\wedge^n} and whose hom spaces are the canonical image of S n 2n 1S^{n_2-n_1} in the hom space Top */(S n 1,S n 2)Top^{\ast/}(S^{n_1},S^{n_2}) (the image under the smash\dashvhom-adjunct of the identity) if n 2>n 1n_2 \gt n_1, and the point otherwise:

𝕊 SeqFreeMod op(F(n 1),F(n 2)) =StdSpheres(S n 1,S n 2) {im(S n 2n 2Top */(S n 1,S n 2)) forn 1n 2 * otherwise. \begin{aligned} \mathbb{S}_{Seq}FreeMod^{op}(F(n_1), F(n_2)) & = StdSpheres(S^{n_1}, S^{n_2}) \\ & \coloneqq \left\{ \array{ im(S^{n_2-n_2} \to Top^{\ast/}(S^{n_1}, S^{n_2})) & for \; n_1 \leq n_2 \\ \ast & otherwise } \right. \end{aligned} \,.

Hence according to lemma 2 sequential spectra are equivalently Top */Top^{\ast/}-enriched functors on StdSpheres:

SeqSpec[StdSpheres,Top */]. SeqSpec \simeq [StdSpheres, Top^{\ast/}] \,.

In this form the statement appears also as (Lydakis 98, prop. 4.3). See also at sequential spectra – As diagram spectra.

Proof

The free 𝕊 Dia\mathbb{S}_{Dia} modules are those of the form y(d)𝕊 Diay(d)\wedge \mathbb{S}_{Dia}, where dDiad \in Dia is an object, and yy denotes the Yoneda embedding. For the case Dia=SeqDia = Seq this means that they are labeled by kk \in \mathbb{N}, their underlying functor F(k)[Seq,Top */]F(k) \in [Seq,Top^{\ast/}] is given by

F(k):n{S nk fornk * otherwise F(k) \;\colon\; n \mapsto \left\{ \array{ S^{n-k} & for\; n \geq k \\ \ast & otherwise } \right.

and the 𝕊 Seq\mathbb{S}_{Seq}-action on these is, expressed as modules over monoidal functors (via this fact about modules with respect to Day convolution) by the canonical identifications

S n 1F(k) n 2=S n 1S n 2kS n 1+n 2k=F(k) n 1+n 2 S^{n_1} \wedge F(k)_{n_2} = S^{n_1} \wedge S^{n_2-k} \overset{}{\longrightarrow} S^{n_1 + n_2 - k} = F(k)_{n_1+ n_2}

whenever n 2kn_2 \geq k, and by the zero morphism otherwise.

Now for RR any other 𝕊 Seq\mathbb{S}_{Seq}-module, with action

S n 1R n 2ρ n 1,n 2R n 1+n 2 S^{n_1} \wedge R_{n_2} \overset{\rho_{n_1,n_2}}{\longrightarrow} R_{n_1 + n_2}

then a homomorphism of 𝕊 Seq\mathbb{S}_{Seq}-modules ϕ:F(k)R\phi \colon F(k)\to R has components ϕ n:S nkR n\phi_{n}\colon S^{n-k} \to R_n fitting into commuting diagrams of the form

S n 1S n 2k S n 1+n 2k (id,ϕ n 2) ϕ n 1+n 2 S n 1R n 2 ρ n 1,n 2 R n 1+n 2 \array{ S^{n_1} \wedge S^{n_2 - k} &\overset{\simeq}{\longrightarrow}& S^{n_1 + n_2 - k} \\ {}^{\mathllap{(id,\phi_{n_2})}}\downarrow && \downarrow^{\mathrlap{\phi_{n_1+n_2}}} \\ S^{n_1} \wedge R_{n_2} &\overset{\rho_{n_1,n_2}}{\longrightarrow}& R_{n_1+n_2} }

for n 2kn_2 \geq k. By the fact that the top morphism here is an isomorphism, all the components ϕ nk\phi_{n \geq k} are uniquely fixed by the component ϕ k\phi_k as

ϕ ngeqk=ρ nk,kϕ k \phi_{n geq k} = \rho_{n-k,k} \circ \phi_k

(and ϕ n<k=0\phi_{n \lt k} = 0). Of course this just confirms the free property of free spectra: morphisms of 𝕊 Seq\mathbb{S}_{Seq}-modules F(k)RF(k) \longrightarrow R are equivalent to morphisms in [Seq,Top */][Seq,Top^{\ast/}] from y(k)y(k) to RR, which by the Yoneda lemma form the space R kTop */R_k \in Top^{\ast/}.

Specialized to RR a free spectrum itself this verifies that the hom-spaces between free 𝕊 Seq\mathbb{S}_{Seq}-modules are as claimed:

𝕊 SeqFreeMod(F(k 2),F(k 1))F(k 1) k 2{S k 2k 1 fork 2k 1 * otherwise. \mathbb{S}_{Seq}FreeMod(F(k_2),F(k_1)) \simeq F(k_1)_{k_2} \simeq \left\{ \array{ S^{k_2-k_1} & for \; k_2 \geq k_1 \\ \ast & otherwise } \right. \,.

Of course this is just the defining free property. But now comparison with the above commuting square for the case n 1=k 1n_1 = k_1 and n 2=k 2n_2 = k_2 shows that composition in 𝕊 SeqFreeMod op\mathbb{S}_{Seq} FreeMod^{op} is indeed the composition in Top */Top^{\ast/} under the identification of F(k)F(k) with S kS^k and of the above hom-space S k 2k 1S^{k_2-k_1} with its image under S k 2k 1Top */(S k 1,S k 2)S^{k_2-k_1} \to Top^{\ast/}(S^{k_1},S^{k_2}).

From lemma 2 we immediately get the following:

Proposition

The horizontal adjunctions in remark 3 lift to adjunctions between categories of modules, such as to give a commuting diagram of adjunctions as follows:

OrthSpec(Top) SymSpec(Top) SeqSpec(Top) 𝕊Mod r orth *orth ! 𝕊 OrthMod r sym *sym ! 𝕊 SymMod r seq *seq ! 𝕊 SeqMod r U F U F U F U F [Top fin */,Top */] orth *orth ! [Orth,Top */] sym *sym ! [Sym,Top */] seq *seq ! [Seq,Top */]. \array{ && OrthSpec(Top) && SymSpec(Top) && SeqSpec(Top) \\ && \downarrow^{\mathrlap{\simeq}} && \downarrow^{\mathrlap{\simeq}} && \downarrow^{\mathrlap{\simeq}} \\ \mathbb{S} Mod_r &\stackrel{\overset{orth_!}{\longleftarrow}}{\underset{orth^\ast}{\longrightarrow}}& \mathbb{S}_{Orth} Mod_r &\stackrel{\overset{sym_!}{\longleftarrow}}{\underset{sym^\ast}{\longrightarrow}}& \mathbb{S}_{Sym} Mod_r &\stackrel{\overset{seq_!}{\longleftarrow}}{\underset{seq^\ast}{\longrightarrow}}& \mathbb{S}_{Seq} Mod_r \\ {}^{\mathllap{U}}\downarrow \uparrow^{\mathrlap{F}} && {}^{\mathllap{U}}\downarrow \uparrow^{\mathrlap{F}} && {}^{\mathllap{U}}\downarrow \uparrow^{\mathrlap{F}} && {}^{\mathllap{U}}\downarrow \uparrow^{\mathrlap{F}} \\ [Top_{fin}^{\ast/}, Top^{\ast/}] &\stackrel{\overset{orth_!}{\longleftarrow}}{\underset{orth^\ast}{\longrightarrow}}& [Orth, Top^{\ast/}] &\stackrel{\overset{sym_!}{\longleftarrow}}{\underset{sym^\ast}{\longrightarrow}}& [Sym, Top^{\ast/}] &\stackrel{\overset{seq_!}{\longleftarrow}}{\underset{seq^\ast}{\longrightarrow}}& [Seq, Top^{\ast/}] } \,.

(MMSS 00, prop. 3.4 with construction 2.1)

We consider now, prop. 4 below, “strict” model category structures on the above categories of spectra, which regard spectra only as diagrams of topological spaces, ignoring the fact that it is not the degreewise homotopy groups but the stable homotopy groups that are to be invariants of stable homotopy types (this is def. 5 below with an extra subtlety in the case of symmetric spectra, see prop. 2 below). Incorporating the latter is accomplished by a left Bousfield localizatiob of the strict model structures to the genuine stable model structures below.

Proposition

Write [Seq,Top Quillen */][Seq, Top^{\ast/}_{Quillen}] for the model category structure on \mathbb{N}-parameterized sequences of pointed topological spaces, whose weak equivalences, fibrations and cofibrations are degreewise those of the classical model structure on pointed topological spaces. (This is equivalently the injective as well as the projective model structure on functors over the discrete category \mathbb{N}).

Then for each of the categories in the diagram of prop. 3, say that the strict model structure on it is the transferred model structure from [Seq,Top Quillen */][Seq, Top^{\ast/}_{Quillen}] along the composite adjunction connecting the two in that diagram, so that we get a diagram of Quillen adjunctions

OrthSpec(Top) strict SymSpec(Top) strict SeqSpec(Top) strict 𝕊Mod strict orth *orth ! 𝕊 OrthMod strict sym *sym ! 𝕊 SymMod strict seq *seq ! 𝕊 SeqMod strict U F U F U F U F [Top fin */,Top */] strict orth *orth ! [Orth,Top */] strict sym *sym ! [Sym,Top */] strict seq *seq ! [Seq,Top Quillen */]. \array{ && OrthSpec(Top)_{strict} && SymSpec(Top)_{strict} && SeqSpec(Top)_{strict} \\ && \downarrow^{\mathrlap{\simeq}} && \downarrow^{\mathrlap{\simeq}} && \downarrow^{\mathrlap{\simeq}} \\ \mathbb{S} Mod_{strict} &\stackrel{\overset{orth_!}{\longleftarrow}}{\underset{orth^\ast}{\longrightarrow}}& \mathbb{S}_{Orth} Mod_{strict} &\stackrel{\overset{sym_!}{\longleftarrow}}{\underset{sym^\ast}{\longrightarrow}}& \mathbb{S}_{Sym} Mod_{strict} &\stackrel{\overset{seq_!}{\longleftarrow}}{\underset{seq^\ast}{\longrightarrow}}& \mathbb{S}_{Seq} Mod_{strict} \\ {}^{\mathllap{U}}\downarrow \uparrow^{\mathrlap{F}} && {}^{\mathllap{U}}\downarrow \uparrow^{\mathrlap{F}} && {}^{\mathllap{U}}\downarrow \uparrow^{\mathrlap{F}} && {}^{\mathllap{U}}\downarrow \uparrow^{\mathrlap{F}} \\ [Top_{fin}^{\ast/}, Top^{\ast/}]_{strict} &\stackrel{\overset{orth_!}{\longleftarrow}}{\underset{orth^\ast}{\longrightarrow}}& [Orth, Top^{\ast/}]_{strict} &\stackrel{\overset{sym_!}{\longleftarrow}}{\underset{sym^\ast}{\longrightarrow}}& [Sym, Top^{\ast/}]_{strict} &\stackrel{\overset{seq_!}{\longleftarrow}}{\underset{seq^\ast}{\longrightarrow}}& [Seq, Top^{\ast/}_{Quillen}] } \,.

Equivalently, under the identification

𝕊 diaMod[𝕊 diaFreeMod op,Top */] \mathbb{S}_{dia} Mod \simeq [\mathbb{S}_{dia} FreeMod^{op}, Top^{\ast/}]

of lemma 2, the strict model structure is the projective model structure on enriched functors (Piacenza 91) see hereodel structure on topological spaces#ModelStructureOnTopEnrichedFunctors).

(MMSS 00, def. 6.1)

Proof

To see that indeed all the adjunctions here are Quillen adjunctions, use that by prop. 1 and lemma 2 all right adjoints are given by restrictions of sites. By definition of the strict model structures and transferred model structures their fibrations and acyclic fibrations are objectwise so, and hence are preserved by these restriction functors.

Remark

The model structure

𝕊 seqMod strictSeqSpec(Top) strict \mathbb{S}_{seq} Mod_{strict} \simeq SeqSpec(Top)_{strict}

in prop. 4 is the strict model structure on topological sequential spectra.

The strict model structures of prop. 4 present the homotopy theory of the given diagrams of homotopy types, hence a homotopy theory of pre-spectra. To obtain from this the genuine stable homotopy theory of genuine spectra we need to restrict this to Omega-spectra, in the following sense.

Definition

For any of the four categories of spectra in prop. 3, we say (with notation from def. 4) that:

  1. an object XX is an Omega-spectrum if seq *Xseq^\ast X is an Omega spectrum in the standard sense of sequential spectra;

  2. a morphism ff is a stable weak homotopy equivalence if seq *(f)seq^\ast(f) is a stable weak homotopy equivalence in the standard sense of sequential spectra (an isomorphism on all stable homotopy groups of sequential spectra);

  3. a morphism ff is a stable equivalence if for all Omega-spectra XX in the above sense, the morphism [f,E] strict[f,E]_{strict} is a bijection (where [,E] strict[-,E]_{strict} is the hom-functor of the homotopy category of the strict model structure of prop. 4.

(MMSS00, def. 8.3 with the notation from p. 21)

The following proposition says that def. 5 makes sense, in that if the left Bousfield localization of the strict model structures at the stable equivalences exists (which we check below) then it indeed localizes the homotopy theory to the Omega-spectra.

Proposition

The weak equivalences in the strict model structure (prop. 4) and the stable equivalences of def. 5 are related as follows:

  1. Every weak equivalence with respect to the strict model structure is a stable equivalence.

  2. Every stable equivalence between Omega-spectra (def. 5) is a weak equivalence in the strict model structure.

(MMSS 00, lemma 8.11)

Proof

The first statement follows by def. 5 since weak equivalences become isomorphisms in the homotopy category.

For the second statement, let f:XYf \colon X \to Y be a stable equivalence between Omega-spectra. Then by definition, in particular

[f,X] strict:[Y,X] strict[X,X] strict [f,X]_{strict} \;\colon\; [Y,X]_{strict} \longrightarrow [X,X]_{strict}

is a bijection. Therefore the pre-image of [id X][X,X] strict[id_X] \in [X,X]_{strict} is an inverse to ff in the homotopy category of the strict model structure. Hence ff represents an isomorphism in the strict homotopy category and is hence a weak equivalence in the strict model structure.

Free spectra

This is a technical section with discussion of certain free objects in the categories of spectra of prop. 2. The discussion here provides technical lemmas that are used in the proof of the stable model structures below, the proof of the Quillen equivalences between them further below and then the proof that the model structure is monoidal.

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. 9 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 6 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. The fine analysis of this phenomenon is crucial in the proof of the Quillen equivalences between the stable model structures on the four models of spectra (theorem 3).

Definition

For dia{Top fin */,Orth,Sym,Seq}dia \in \{Top^{\ast/}_{fin}, Orth, Sym, Seq\} and for each nn \in \mathbb{N}, the functor

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

that sends a structured spectrum (notation as in def. 4) to the nnth component space of its underlying sequential spectrum has 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.

The adjunction units in the sequence of adjunctions equip these with canonical natural transformations

F n Seqf n SeqF n Symf n SymF n Orthf n OrthF n Top fin */. F_n^{Seq} \stackrel{f_n^{Seq}}{\longrightarrow} F_n^{Sym} \stackrel{f_n^{Sym}}{\longrightarrow} F_n^{Orth} \stackrel{f_n^{Orth}}{\longrightarrow} F_n^{Top^{\ast/}_{fin}} \,.

(MMSS 00, section 8)

Lemma

Under the equivalence of lemma 2

  1. the free spectrum on KTop */K \in Top^{\ast/} is

    F n diaK=𝕊 diaFreeMod(,y(n)𝕊 dia)K[𝕊 diaFreeMod op,Top */]𝕊 diaMod, F^{dia}_n K \;=\; \mathbb{S}_{dia}FreeMod( - , y(n)\wedge \mathbb{S}_{dia} ) \wedge K \;\in\; [ \mathbb{S}_{dia} FreeMod^{op}, Top^{\ast/}] \;\stackrel{\simeq}{\to}\; \mathbb{S}_{dia}Mod \,,

    where y(n)y(n) denotes the functor represented by dia(n)dia(n) (i.e. by nn if dia=Seqdia = Seq, by {1,,n}\{1,\cdots,n\} if dia=Symdia = Sym, by n\mathbb{R}^n if dia=Orthdia = Orth and by S nS^n if dia=Top fin */dia = Top^{\ast/}_{fin}).

  2. On the free 𝕊 dia\mathbb{S}_{dia}-module on eDia opy[Dia,Top */]e \in Dia^{op} \stackrel{y}{\hookrightarrow} [Dia, Top^{\ast/}] this takes the value

(F n diaK)(e)=lime 1e 2eDia(n,e 1)𝕊 dia(e 2)K. (F^{dia}_n K)(e) \;=\; \underset{\underset{e_1 \otimes e_2 \to e}{\longrightarrow}}{\lim} Dia(n,e_1)\wedge \mathbb{S}_{dia}(e_2) \wedge K \,.

(MMSS00, p. 7 with theorem 2.2)

Proof

Generally, for 𝒞\mathcal{C} a VV-enriched symmetric monoidal category, and for c𝒞c\in \mathcal{C} an object, then there is an adjunction

[𝒞,V]() cy(d)()V [\mathcal{C}, V] \stackrel{\overset{y(d)\otimes(-)}{\longleftarrow}}{\underset{(-)_c}{\longrightarrow}} V

whose characterizing natural isomorphism is the combination of that of tensoring with the Yoneda embedding:

[𝒞,V](y(d)K,F)V(K,[𝒞,V](y(d),F))V(K,F(d)). [\mathcal{C},V]( y(d) \otimes K, F ) \simeq V( K, [\mathcal{C},V]( y(d), F ) ) \simeq V(K, F(d)) \,.

Specializing this to our case with V=Top */V = Top^{\ast/} and 𝒞=𝕊 diaFreeMod op\mathcal{C} = \mathbb{S}_{dia} FreeMod^{op} yields the first statement. The second follows from similar yoga, see the formula at Day convolution – Modules.

Proposition

Explicitly, the free spectra according to def. 6, look as follows:

For sequential spectra: (F n SeqK) qKS qn(F^{Seq}_n K)_q \simeq K \wedge S^{q-n};

for symmetric spectra: (F n SymK) qΣ(q) + Σ(qn)KS qn(F^{Sym}_n K)_q \simeq \Sigma(q)_+ \wedge_{\Sigma(q-n)} K \wedge S^{q-n}.

for orthogonal spectra: (F n OrthK) qO(q) + O(qn)KS qn(F^{Orth}_n K)_q \simeq O(q)_+ \wedge_{O(q-n)} K \wedge S^{q-n}.

In particular:

  1. F 0K=Σ KF_0 K = \Sigma^\infty K;

  2. F n SeqS nF_n^{Seq}S^n is like the suspension spectrum of the point (the standard sequential sphere spectrum) but with its first nn components simply removed.

(e.g. Schwede 12, example 3.20)

Proof

By working out the formula in item 2 of lemma 3. In the sequential case (Dia=Seq)(Dia = Seq) there exists a morphism k 1k 2k 3k_1 \otimes k_2 \to k_3 only if k 1+k 2=k 3k_1 + k_2 = k_3 and then there is a unique such. Hence here the colimit in the formula becomes a coproduct and we find

(F n SeqK)(q) e 1+e 2=qSeq(n,e 1){S 0 ifn=e 1 * otherwise𝕊 dia(e 2)K S qnK. \begin{aligned} (F^{Seq}_n K)(q) & \simeq \underset{e_1+e_2 = q}{\coprod} \underset{\simeq\left\{ \array{S^0 & if\, n = e_1 \\ \ast & otherwise}\right.}{\underbrace{Seq(n,e_1)}} \wedge \mathbb{S}_{dia}(e_2) \wedge K \\ & \simeq S^{q-n}\wedge K \end{aligned} \,.

In the symmetric case (Dia=SymDia = Sym) the formula is similar except that S 0=Seq(q,q) +S^0 = Seq(q,q)_+ is replaced by Sym(q,q) +=Σ(q) +Sym(q,q)_+ = \Sigma(q)_+ and the colimit goes over the automorphisms that fix qnq-n elements, thereby producing the partial smash tensor shown in the statement. Analogously for the orthogonal case (Dia=OrthDia = Orth).

One use of free spectra, important in the verification of the stable model structures below and in the dicussion of the stable equivalences further below, 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

such that for every X𝕊 diaModX\in \mathbb{S}_{dia} Mod the operation λ n *\lambda_n^\ast of precomposition with λ n\lambda_n 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. 4).

Proof

Since all prescribed morphisms in the diagram are natural transformations, this is in fact a diagram of copreheaves 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. 6 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. 6 and the second comes from the adjunction units according to def. 6.

(MMSS 00, def. 8.4, Schwede 12, example 4.26)

Lemma

The morphisms of def. 7 are those whose existence is asserted by prop. 4.

(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 prop. 6, 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 component f 0:S 0X 0f_0 \colon S^0 \to X_0 (that’s of course the 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 \,.

The following property of the maps from def. 7 to be or not to be stable weak homotopy equivalences will be the key technical fact that implies (below) that stable equivalences of spectra are or are not the same as stable weak homotopy equivalences.

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. 7 are

  1. stable equivalences, according to def. 5, for all four cases of spectra, Dia{Top */,Orth,Sym,Seq}{Dia} \in \{Top^{\ast/}, Orth, Sym, Seq\};

  2. stable weak homotopy equivalences, according to def. 5, for sequential spectra, symmetric spectra and pre-excisive functors Dia{Top */,Orth,Seq}{Dia} \in \{Top^{\ast/}, Orth, Seq\};

  3. 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

The first statement is an immediate consequence of lemma 5.

The other two statements follow from inspection of the explicit form of the maps, via prop. 6, in 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 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

For A,BTop */A, B \in Top^{\ast/} and for k,k,\ell \in \mathbb{N}, then the symmetric monoidal smash product of spectra, def. 3, applied to the corresponding free spectra from def. 6 relates to the plain smash product of pointed topological spaces via natural isomorphisms

(F kA) 𝕊 dia(F B)F k+(AB). (F_k A)\wedge_{\mathbb{S}_{dia}} (F_\ell B) \simeq F_{k+\ell}(A\wedge B) \,.

(MMSS 00, lemma 1.8, lemma 21.3)

Proof

Consider the following sequence of natural isomorphisms

[𝕊 diaFreeMod op,Top */]((F kA) 𝕊 dia(F B),Z) [𝕊 diaFreeMod op×𝕊 diaFreeMod op,Top */]((F kA)˜(F B),Z) Top */(AB,F k+) [𝕊 diaFreeMod op,Top */](F k+(AB),Z), \begin{aligned} [\mathbb{S}_{dia} FreeMod^{op},Top^{\ast/}]((F_k A)\wedge_{\mathbb{S}_{dia}} (F_\ell B), Z) & \simeq [\mathbb{S}_{dia} FreeMod^{op}\times \mathbb{S}_{dia} FreeMod^{op}, Top^{\ast/}]((F_k A)\tilde \wedge (F_\ell B), Z \circ \wedge) \\ & \simeq Top^{\ast/}( A\wedge B, F_{k+\ell}) \\ & \simeq [\mathbb{S}_{dia} FreeMod^{op},Top^{\ast/}]( F_{k+\ell}(A \wedge B), Z ) \end{aligned} \,,

where we used the adjoint characterization (here) of the Day convolution. Since this is natural in ZZ, the Yoneda lemma implies the claim.

Lemma

The symmetric monoidal smash product of spectra of the free spectrum constructions (def. 6) 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 7 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.

Part II. Model categories of diagram spectra

We now discuss equipping the diagram categories of part I with model category structures, each presenting the stable homotopy theory (the stable (infinity,1)-category of spectra), and how the system of adjunctions between these categories becomes a system of Quillen equivalences between these model structures.

5.-10.) Model structures on plain spectra

Here we discuss model structures for plain spectra, below we discuss model structures for ring spectra and module spectra.

The stable model structures

Theorem

The left Bousfield localization of the strict model structures of prop. 4 at the stable equivalences of def. 5 exists, to be called the stable model structures.

Explicitly, there is a model structure 𝕊 diaMod stable\mathbb{S}_{dia} Mod_{stable} whose

  • cofibrations are those of the strict model structure (prop. 4);

  • weak equivalences are the stable equivalences (def. 5)

and the identity functors constitute a Quillen adjunction of the form

𝕊 diaMod stableidid𝕊 diaMod strict. \mathbb{S}_{dia} Mod_{stable} \stackrel{\overset{id}{\longleftarrow}}{\underset{id}{\longrightarrow}} \mathbb{S}_{dia} Mod_{strict} \,.

Moreover 𝕊 diaMod stable\mathbb{S}_{dia} Mod_{stable} is a

  1. compactly generated model category;

  2. proper model category;

  3. Top Quillen */Top^{\ast/}_{Quillen}-enriched model category.

Specifically, this specializes to

(MMSS00, theorem 9.2)

We give the proof of theorem 1 below, it involves the following definitions and lemmas.

The generating cofibrations and acylic cofibrations are going to be the those induced via tensoring of representables from the classical model structure on topological spaces (giving the strict model structure), together with an additional set of morphisms to the generating acylic cofibrations that will force fibrant objects to be Omega-spectra. To that end we need the following little preliminary.

Remark

By construction, the morphisms λ n\lambda_n of lemma 4 are stable equivalences according to def. 5.

We need to in addition resolve them by suitable cofibrations:

Definition

For nn \in \mathbb{N} let

λ n:F n+k nCyl(λ n)def.retr.F nS 0 \lambda_n \colon F_{n+} \stackrel{k_n}{\longrightarrow} Cyl(\lambda_n) \stackrel{def.retr.}{\longrightarrow} F_n S^0

be a factorization of the morphism λ n\lambda_n of lemma 4 and def. 7 through a strict cofibration followed by a strict weak equivalence (e.g. through its mapping cylinder followed by a deformation retraction (see here)).

With this we may state the classes of morphisms that are going to be shown to be the classes of generating (acyclic) cofibrations for the stable model structures:

Definition

Recall the sets

I{S n1D n} n I \coloneqq \{S^{n-1} \hookrightarrow D^n\}_{n \in \mathbb{N}}
J{D nD n×I} n J \coloneqq \{D^n \hookrightarrow D^n \times I\}_{n \in \mathbb{N}}

of generating cofibrations and generating acyclic cofibrations, respectively, of the classical model structure on topological spaces.

Write

FI{y(x)i +} x𝕊 DiaFreeModiI F I \coloneqq \{ y(x) \otimes i_+ \}_{{x \in \mathbb{S}_{Dia} FreeMod} \atop {i \in I}}

for the class of free spectra, def. 6, on the class II above, which by lemma 3 is equivalently the set of morphisms arising as the tensoring with a topological generating cofibration of a representable over the site 𝕊 diaFreeMod\mathbb{S}_{dia} FreeMod (the site for 𝕊 diaMod\mathbb{S}_{dia}Mod from lemma 2).

Similarly, write

FJ{y(x)j +} x𝕊 DiaFreeModjJ, F J \coloneqq \{ y(x) \otimes j_+ \}_{{x \in \mathbb{S}_{Dia}FreeMod} \atop {j \in J}} \,,

for the set of morphisms arising as the tensoring of a representable with a generating acyclic cofibration of the classical model structure on topological spaces (with basepoint adjoined).

Finally write

KFJ{k ni +} niI K \coloneqq F J \;\sqcup\; \{ k_n \Box i_+ \}_{{n \in \mathbb{N}} \atop {i \in I}}

for the disjoint union of FJF J with the pushout products of the resolved maps k nk_n from def. 8 with the elements in II.

(MMSS 00, def. 9.3)

Proposition

The sets FIF I and FJF J from def. 9 (disregarding the set KK there) are, respectively sets of generating cofibrations and generating acyclic cofibrations for the strict model structure 𝕊 DiaMod strict\mathbb{S}_{Dia}Mod_{strict} (prop. 4).

Proof

By prop. 4 the strict model structure is equivalently the projective pointed model structure on topologically enriched functors

𝕊 DiaMod strict[𝕊 DiaFreeMod op,Top */] proj. \mathbb{S}_{Dia}Mod_{strict} \simeq [\mathbb{S}_{Dia}FreeMod^{op}, Top^{\ast/}]_{proj} \,.

With this the statement follows by the proof of this theorem.

Lemma

Every element in KK (def. 9) is both:

  1. a cofibration with respect to the strict model structure (prop. 4);

  2. a stable equivalence (def. 5).

Proof

First regarding strict cofibrations: By the Yoneda lemma, the elements in FJF J have right lifting property against the strict fibrations, hence in particular they are strict cofibrations. Moreover, by Joyal-Tierney calculus, k ni +k_n \Box i_+ has left lifting against any acyclic strict fibration ff precisely if k nk_n has left lifting against f if^i. By 𝕊 diaMod strict\mathbb{S}_{dia} Mod_{strict} behaving like a TopTop-enriched model category for one argument a relative CW-complex, the latter is still a strict acyclic fibration. Since k nk_n by construction is a strict cofibration, the lifting follows and hence also k ni +k_n \Box i_+ is a strict cofibration.

Regarding stable equivalences: The morphisms in FJF J by design are strict weak equivalences, hence they are in particular stable equivalences. Similarly, the morphisms k nk_n by construction, by two-out-of-three and by remark 5 are stable equivalences. Hence the derived hom (…expand…) out of k ni +k_n \Box i_+ is the homotopy pullback of a weak equivalence, hence is a weak equivalence, hence on the homotopy category an iso.

The point of the class KK in def. 9 is to make the following true:

Lemma

A morphism f:XYf \colon X \to Y in 𝕊 diaMod\mathbb{S}_{dia} Mod is a KK-injective morphism (for KK from def. 9) precisely if

  1. it is a fibration in the strict model structure (hence degreewise a fibration)

  2. for all nn \in \mathbb{N} the commuting squares of structure map compatibility on the underlying sequential spectra

    X n σ˜ ΩX n+1 Y n σ˜ ΩY n+1 \array{ X_n &\overset{\tilde\sigma}{\longrightarrow}& \Omega X_{n+1} \\ \downarrow && \downarrow \\ Y_n &\underset{\tilde \sigma}{\longrightarrow}& \Omega Y_{n+1} }

    exhibit homotopy pullbacks.

(MMSS 00, prop. 9.5)

Proof

By prop 7, lifting against FJF J alone characterizes strict fibrations, hence degreewise fibrations. Lifting against the remaining pushout product morphism k ni +k_n \Box i_+ is, by Joyal-Tierney calculus, equivalent to left lifting i +i_+ against the dual pullback product of f k nf^{k_n}, which means that f k nf^{k_n} is a weak homotopy equivalence. But by construction (lemma 4) f k nf^{k_n} is the comparison morphism into the homotopy pullback under consideration.

Corollary

The KK-injective objects (for KK from def. 9) are precisely the Omega-spectra, def. 5.

Lemma

A morphism in 𝕊 diaMod\mathbb{S}_{dia}Mod which is both

  1. a stable equivalence (def. 5);

  2. a KK-injective morphisms (with respect to KK from def. 9)

is an acyclic fibration in the strict model structure of prop. 4, hence is degreewise a weak homotopy equivalence and Serre fibration of topological spaces;

(MMSS 00, corollary 9.8)

Proof

Let f:XBf\colon X \to B be both a stable equivalence as well as a KK-injective morphism. Since KK contains, by prop. 7, the generating acyclic cofibrations for the strict model structure of prop. 4, ff is in particular a strict fibration, hence a degreewise fibration. Therefore the fiber FF of ff is its homotopy fiber in the strict model structure.

We now want to claim that:

It follows that F*F \to \ast is a stable equivalence.

This needs some explanation:

Consider for any E𝕊 diaModE \in \mathbb{S}_{dia}Mod the cofiber sequence

[B,E] strictp *[X,E] strict[hocof(p),E] strict[ΣB,E] strictΣp *[ΣX,E] strict[B,E]_{strict} \overset{p^\ast}{\longrightarrow} [X,E]_{strict} \overset{}{\longrightarrow} [hocof(p),E]_{strict} \longrightarrow [\Sigma B, E]_{strict} \overset{\Sigma p^\ast}{\longrightarrow}[\Sigma X,E]_{strict}

This kind of sequence is long exact for every pointed model category, not necessarily stable. Then let EE be any Omega-spectrum. By assumption it follows then that p *p^\ast and Σp *\Sigma p^\ast are isomorphisms, so that exactness implies that [hocof(p),E] strict=0[hocof(p),E]_{strict} = 0 for all Omega-spectra EE.

Now use that on underlying sequential spectra there is a stable weak homotopy equivalence hocof(p)Σhofib(p)=ΣFhocof(p) \longrightarrow \Sigma hofib(p) = \Sigma F (prop.). Since suspension is preserved by passing to underlying sequential spectra, and since stable homotopy groups by definition are those of the underlying sequential spectra, and since stable weak equivalences are in particular stable equivalences (prop. 8 below) it follows that [ΣF,E] strict[\Sigma F, E]_{strict} for all Omega-spectra EE, hence [F,ΩE] strict=0[F, \Omega E]_{strict} = 0 for all Omega-spectra.

We are to conclude that hence F*F \to \ast is a stable equivalence. But to conclude this we now need to know that every Omega-spectrum in 𝕊 diaMod\mathbb{S}_{dia}Mod is in the image under Ω\Omega of an Omega-spectrum, up to strict equivalence. Indeed one shows that there is a shift functor shsh on structured spectra and that EΩshEE \overset{}{\longrightarrow} \Omega sh E is degreewise a weak homotopy equivalence.

Observe also that FF, being the pullback of a KK-injective morphisms (by the standard closure properties) is a KK-injective object, so that by corollary 1 FF is an Omega-spectrum. Together this implies with prop. 5 that F*F \to \ast is a weak equivalence in the strict model structure, hence degreewise a weak homotopy equivalence. From this the long exact sequence of homotopy groups implies that π 1(f n)\pi_{\bullet \geq 1}(f_n) is a weak homotopy equivalence for all nn and for each homotopy group in positive degree.

To infer from this the remaining case that also π 0(f 0)\pi_0(f_0) is an isomorphism, observe that, by assumption of KK-injectivity, lemma 10 gives that f nf_n is a homotopy pullback (in topological spaces) of Ω(f n+1)\Omega (f_{n+1}). But, by the above, Ω(f n+1)\Omega (f_{n+1}) is a weak homotopy equivalence, since π (Ω())=π +1()\pi_\bullet(\Omega(-)) = \pi_{\bullet+1}(-). Therefore f nf_n is the homotopy pullback of a weak homotopy equivalence and hence itself a weak homotopy equivalence.

Lemma

For KK from def. 9 the retracts of KK-relative cell complexes are precisely the morphisms which are

  1. stable equivalences (def. 5),

  2. as well as cofibrations with respect to the strict model structure of prop. 4.

(MMSS 00, prop. 9.9 (i))

Proof

Since all elements of KK are stable equivalences and strict cofibrations by lemma 9, it follows that every retract of a relative KK-cell complex has the same property.

In the other direction, if ff is a stable equivalence and strict cofibration, by the small object argument it factors f:ipf \colon \stackrel{i}{\to}\stackrel{p}{\to} as a relative KK-cell complex ii followed by a KK-injective morphism pp. By the previous statement ii is a stable equivalence, and so by assumption and by two-out-of-three so is pp. Therefore lemma 11 implies that pp is a strict acyclic fibration. But then the assumption that ff is a strict cofibration means that it has the left lifting property against pp, and so the retract argument implies that ff is a retract of the relative KK-cell complex ii.

Corollary

For KK from def. 9 the KK-injective morphisms are precisely those which are injective with respect to the cofibrations of the strict model structure that are also stable equivalences.

(MMSS 00, prop. 9.9 (ii))

Lemma

A morphism in 𝕊 diaMod\mathbb{S}_{dia}Mod is both

  1. a stable equivalence (def. 5)

  2. injective with respect to the cofibrations of the strict model structure that are also stable equivalences;

precisely if it is an acylic fibration in the strict model structure (prop. 4).

(MMSS 00, prop. 9.9 (iii))

Proof

Every acyclic fibration in the strict model structure is injective with respect to strict cofibrations by the strict model structure; and it is a stable equivalence by item 1 of prop. 5.

Conversely, a morphism injective with respect to strict cofibrations that are stable equivalences is a KK-injective morphism by corollary 2, and hence if it is also a stable equivalence then by lemma 11 it is a strict acylic fibration.

Proof

(of theorem 1)

The non-trivial points to check are the two weak factorization systems.

That (cof stableweq stable,fib stable)(cof_{stable}\cap weq_{stable} \;,\; fib_{stable}) is a weak factorization system follows from lemma 12 and the small object argument.

By lemma 13 the stable acyclic fibrations are equivalently the strict acyclic fibrations and hence the weak factorization system (cof stable,fib stablewe stable)(cof_{stable} \;,\; fib_{stable} \cap we_{stable}) is identified with that of the strict model structure (cof strict,fib strictwe strict)(cof_{strict} \;,\; fib_{strict} \cap we_{strict}).

Stable equivalences

Here we discuss that the two concepts of stable equivalences and of stable weak homotopy equivalences in def. 5 actually agree in the cases of a) pre-excisive functors, b) orthogonal spectra and c) sequential spectra, while in the case of symmetric spectra the class of stable equivalences includes but is strictly larger than that of stable weak homotopy equivalences.

This is important in practice, since while the stable equivalences are the weak equivalences in the stable model structure of theorem 1, it is the stable weak homotopy equivalences that are typically more readily identified.

Theorem

In 𝕊Mod\mathbb{S}Mod, 𝕊 OrthMod\mathbb{S}_{Orth} Mod and in 𝕊 SeqMod\mathbb{S}_{Seq} Mod we have for the concepts from def. 5 that

stableweakhomotopyequivalencestableequivalence. stable\;weak\;homotopy\;equivalence \;\Leftrightarrow\; stable \; equivalence \,.

In 𝕊 SymMod\mathbb{S}_{Sym}Mod however we only have

stableweakhomotopyequivalencestableequivalence stable\;weak\;homotopy\;equivalence \;\Rightarrow\; stable \; equivalence

but the reverse implication is false.

(MMSS00, prop. 8.7, prop. 8.8)

We break up this statement below as prop. 8 and prop. 9.

The argument that every stable weak homotopy equivalence is in particular a stable equivalence is fairly formal; this we turn to first in prop. 8 below. The converse statement in prop. 9 however relies on explicit analysis of the class KK of generating acylic cofibrations in def. 9.

Definition

For λ 0:F 1 diaS 1F 0 diaS 0\lambda_0 \colon F^{dia}_1 S^1 \to F^{dia}_0 S^0 from lemma 4, write

(XRX)F(λ 0,X) (X \longrightarrow R X) \coloneqq F(\lambda_0, X)

for the mapping spectrum construction out of λ 1\lambda_1 into XX.

Write

R Xholim(Xλ 0 *(X)RXR(λ 0 *(X))RRXRR(λ 0 *(X))) R^\infty X \coloneqq \underset{\longrightarrow}{ho\lim} \left( X \stackrel{\lambda_0^\ast(X)}{\longrightarrow} R X \stackrel{R(\lambda_0^\ast(X))}{\longrightarrow} R R X \stackrel{R R(\lambda_0^\ast(X))}{\longrightarrow} \cdots \right)

for the homotopy colimit over the resulting sequence of iterations (formed with respec to the strict model structure of prop. 4). Write

r X:XR X r_X \colon X \longrightarrow R^\infty X

for the 0th-component map into the colimit.

Lemma

The functor R R^\infty from def. 10 has the following properties.

  1. for EE an Omega-spectrum according to def. 5, then, by lemma 4, λ 0 *(E)\lambda_0^\ast(E) is weak equivalence in the strict model structure (prop. 4), and hence so is r Er_E;

  2. for f:XYf\colon X \longrightarrow Y a stable weak homotopy equivalence according to def. 5, then R f:R XR YR^\infty f \colon R^\infty X \longrightarrow R^\infty Y is a weak equivalence in the strict model structure.

(MMSS 00, prop. 8.8)

Proof

For the first item, use that the homotopy colimit is represented by the mapping telescope. Then this lemma implies that every element of the homotopy groups of the homotopy colimit is represented at some finite stage. This implies that the telescope of level-wise weak homotopy equivalences is a level-wise weak homotopy equivalence.

For the second item, observe that by the defining adjunctions and by lemma 4 (…) we have

(R nX) qΩ nX(n+q) (R^n X)_q \simeq \Omega^n X(n+q)

and

π q((R X) n)π qn(X). \pi_q((R^\infty X)_n) \simeq \pi_{q-n}(X) \,.
Proposition

In def. 5 every stable weak homotopy equivalence is a stable equivalence.

(MMSS 00, prop. 8.8, following Hovey-Shipley-Smith 00, theorem 3.1.11)

Proof

Let EE be an Omega-spectrum. Then by the first item of lemma 14, for every XX the morphism

[X,r E] strict:[X,E] strict[X,R E] strict [X,r_E]_{strict} \;\colon\; [X,E]_{strict} \longrightarrow [X, R^\infty E]_{strict}

is an isomorphism. Since r ()r_{(-)} is a natural transformation (by def. 10), the naturality squares give a factorization of this morphism as

[X,r E] strict:[X,E] strictR [R X,R E] strict[r X,E] strict[X,R E] strict. [X,r_E]_{strict} \;\colon\; [X,E]_{strict} \stackrel{R^\infty}{\longrightarrow} [R^\infty X, R^\infty E]_{strict} \stackrel{[r_X,E]_{strict}}{\longrightarrow} [X, R^\infty E]_{strict} \,.

Combining this with vertical morphisms as below, which are isomorphisms again by item 1 of lemma 14,

[R X,E] strict [r X,E] strict [X,E] strict [R X,r E] strict [X,r E] strict [X,r E] strict: [X,E] strict R [R X,R E] strict [r X,R E] strict [X,R E] strict \array{ & && [R^\infty X, E]_{strict} &\stackrel{[r_X, E]_{strict}}{\longrightarrow}& [X, E]_{strict} \\ & &\nearrow& {}^{\mathllap{\simeq}}\downarrow^{\mathrlap{[R^\infty X,r_E]_{strict}}} && {}^{\mathllap{\simeq}}\downarrow^{\mathrlap{[X,r_E]_{strict}}} \\ [X,r_E]_{strict} \;\colon\; & [X,E]_{strict} &\stackrel{R^\infty}{\longrightarrow}& [R^\infty X, R^\infty E]_{strict} &\stackrel{[r_X, R^\infty E]_{strict}}{\longrightarrow}& [X, R^\infty E]_{strict} }

exhibits a retraction

id:[X,E] strict[R X,E] strict[X,E] strict, id \colon [X,E]_{strict} \longrightarrow [R^\infty X,E]_{strict} \longrightarrow [X,E]_{strict} \,,

which is natural in XX (that the bottom and right composite is indeed the identity is again the naturality of r ()r_{(-)}). This naturality now implies a retraction of morphisms

id [Y,E] strict: [Y,E] strict [R Y,E] strict [Y,E] strict [f,E] strict [R f,E] strict [f,E] strict id [X,E] strict: [X,E] strict [R X,E] strict [X,E] strict. \array{ id_{[Y,E]_{strict}} \colon & [Y,E]_{strict} &\longrightarrow& [R^\infty Y,E]_{strict} &\longrightarrow& [Y,E]_{strict} \\ & \downarrow^{\mathrlap{[f,E]_{strict}}} && \downarrow^{\mathrlap{[R^\infty f,E]_{strict}}} && \downarrow^{\mathrlap{[f,E]_{strict}}} \\ id_{[X,E]_{strict}} \colon & [X,E]_{strict} &\longrightarrow& [R^\infty X,E]_{strict} &\longrightarrow& [X,E]_{strict} } \,.

Finally, by the second item of lemma 14, the middle vertical morphism here is an isomorphism, hence [f *,E] strict[f^\ast, E]_{strict} is the retract of an iso and hence (here) an isomorphism itself, for all Omega-spectra EE. This means by definition that ff is a stable equivalence.

Now for the converse.

Proposition

In the case Dia{Top */,Orth,Seq}Dia \in \{Top^{\ast/}, Orth, Seq\}, hence for sequential spectra, orthogonal spectra and pre-excisive functors, stable equivalences are stable weak homotopy equivalences (def. 5).

(MMSS 00, p.31-32)

Proof idea

By theorem ref. 1, and by lemmas 11 and 12, every stable weak equivalence factors as a KK-relative cell complex followed by weak equivalence in the strict model structure. Since the latter is degreewise a weak homotopy equivalence it is in particular a stable weak homotopy equivalence. Hence we are reduced to showing that every KK-relative cell complex is a stable weak homotopy equivalence.

Observe now that we know this to be true for the elements of KK itself (def. 9): first of all, the elements in FJF J are retracts of deformation retracts and therefore stable weak homotopy equivalences. Second, the morphisms denoted k nk_n in def. 9, which resolve the maps λ n\lambda_n from def. 7, are stable weak homotopy equivalences by lemma 6. This implies that so are their pushout products k nik_n \Box i.

This completes the proof of theorem 2.

Quillen equivalences between the stable model structures

Theorem

The sequence of Quillen adjunctions between the strict model structures of prop. 4 remain Quillen adjunctions for the stable model structures of theorem 1 and indeed become a sequence of Quillen equivalences

OrthSpec(Top) stable SymSpec(Top) stable SeqSpec(Top) stable 𝕊Mod stable orth * Quorth ! 𝕊 OrthMod stable sym * Qusym ! 𝕊 SymMod stable seq * Quseq ! 𝕊 SeqMod stable. \array{ && OrthSpec(Top)_{stable} && SymSpec(Top)_{stable} && SeqSpec(Top)_{stable} \\ && \downarrow^{\mathrlap{\simeq}} && \downarrow^{\mathrlap{\simeq}} && \downarrow^{\mathrlap{\simeq}} \\ \mathbb{S} Mod_{stable} &\stackrel{\overset{orth_!}{\longleftarrow}}{\underoverset{orth^\ast}{\simeq_{Qu}}{\longrightarrow}}& \mathbb{S}_{Orth} Mod_{stable} &\stackrel{\overset{sym_!}{\longleftarrow}}{\underoverset{sym^\ast}{\simeq_{Qu}}{\longrightarrow}}& \mathbb{S}_{Sym} Mod_{stable} &\stackrel{\overset{seq_!}{\longleftarrow}}{\underoverset{seq^\ast}{\simeq_{Qu}}{\longrightarrow}}& \mathbb{S}_{Seq} Mod_{stable} } \,.

(MMSS 00, section 10)

We give the proof below, after a few preliminaries.

Lemma

The sequence of adjunctions between the categories 𝕊 diaMod\mathbb{S}_{dia}Mod from prop. 3 are Quillen adjunctions with respect to the stable model structures of theorem 1

OrthSpec(Top) stable SymSpec(Top) stable SeqSpec(Top) stable 𝕊Mod stable orth *orth ! 𝕊 OrthMod stable sym *sym ! 𝕊 SymMod stable seq *seq ! 𝕊 SeqMod stable. \array{ && OrthSpec(Top)_{stable} && SymSpec(Top)_{stable} && SeqSpec(Top)_{stable} \\ && \downarrow^{\mathrlap{\simeq}} && \downarrow^{\mathrlap{\simeq}} && \downarrow^{\mathrlap{\simeq}} \\ \mathbb{S} Mod_{stable} &\stackrel{\overset{orth_!}{\longleftarrow}}{\underoverset{orth^\ast}{}{\longrightarrow}}& \mathbb{S}_{Orth} Mod_{stable} &\stackrel{\overset{sym_!}{\longleftarrow}}{\underoverset{sym^\ast}{}{\longrightarrow}}& \mathbb{S}_{Sym} Mod_{stable} &\stackrel{\overset{seq_!}{\longleftarrow}}{\underoverset{seq^\ast}{}{\longrightarrow}}& \mathbb{S}_{Seq} Mod_{stable} } \,.

(MMSS 00, lemma 10.1)

Proof

By lemma 12 the stable fibrations are equivalently the KK-injective morphisms. By lemma 10 these are characterized by data that is preserved by right Quillen functors with respect to the strict model structure. Moreover by lemma 13 the stable acyclic fibrations are equivalently the strict acyclic fibrations, which are of course also preserved by right Quillen functors for the strict model structure. Therefore the statement follows with prop. 4.

Proof idea

(of theorem 3)

With lemma 15 it is sufficient to show that all the total derived functors are adjoint equivalences. By two-out-of-three for Quillen equivalences, it is sufficient to show this for all the (composite) adjunctions whose right adjoint does not point to 𝕊 SymMod\mathbb{S}_{Sym}Mod.

In these cases, theorem 2 implies that the right adjoint functor preserves and reflects weak equivalence (a morphism in its domain is a stable equivalence precisely if its image is).

In such a case, for checking a Quillen equivalence it is sufficient to check that the adjunction unit is a weak equivalence on all cofibrant objects (…citation…).

Since both adjoints in the present case preserve colimits, tensoring with Top */Top^{\ast/} and the homotopy lifting property, and since (…)

(…)

(…)

11.-16.) Model structures on ring spectra and module spectra

Monoidal model structure

Theorem

The stable model structures from theorem 1 on the categories of modules from prop. 2, remark 2

𝕊Mod r[Top fin */,Top */] \mathbb{S}Mod_r \simeq [Top_{fin}^{\ast/}, Top^{\ast/}]
𝕊 OrthMod rOrthSpec(Top) \mathbb{S}_{Orth} Mod_r \simeq OrthSpec(Top)
𝕊 SymMod rSymSpec(Top) \mathbb{S}_{Sym} Mod_r \simeq SymSpec(Top)

are compatible with their monoidal category structure given by the symmetric monoidal smash product of spectra \wedge of def. 3, in that (𝕊 diaMod stable, 𝕊 dia)(\mathbb{S}_{dia} Mod_{stable}, \wedge_{\mathbb{S}_{dia}}) in these cases

  1. is a stable model category;

  2. satisfying the monoid axiom in a monoidal model category.

(MMSS 00, theorem 12.1 (iii) with prop. 12.3)

We give the proof below (…) after a sequence of lemmas.

Lemma

The pushout product of two cofibrations in 𝕊 diaMod stable\mathbb{S}_{dia}Mod_{stable} is again a cofibration.

Proof

A general abstract fact about pushout-products (Hovey-Shipley-Smith 00, prop. 5.3.4, see here) is that for I 1,I 2I_1, I_2 two classes of morphisms in a closed symmetric monoidal category with finite limits and finite colimits, and writing I iCofI_i Cof for their saturated classes, then under pushout-product \Box:

(I 1Cof)(I 2Cof)(I 1I 2)Cof. (I_1 Cof) \Box (I_2 Cof) \subset (I_1 \Box I_2) Cof \,.

Since the cofibrations of the stable model structure, theorem 1, are elements in

Cof stable=(FI)Cof Cof_{stable} = (F I) Cof

with FIF I the class of free spectra on the class of generating cofibrations II of the classical model structure on topological spaces, def. 9, this implies in the present case that

Cof stableCof stable(FIFI)Cof. Cof_{stable} \Box Cof_{stable} \subset (F I \Box F I) Cof \,.

Now lemma 8 implies that

FIFI=F(II)=FI F I \Box F I = F(I \Box I) = F I

and hence the claim follows.

Lemma

Let Y𝕊 diaMod stableY \in \mathbb{S}_{dia} Mod_{stable} be cofibrant. Then the smash product of spectra-functor (def. 3)

X 𝕊 dia():𝕊 diaMod𝕊 diaMod X \wedge_{\mathbb{S}_{dia}}(-) \;\colon\; \mathbb{S}_{dia} Mod \longrightarrow \mathbb{S}_{dia} Mod

preserves stable weak homotopy equivalences as well as stable equivalences (def. 5).

(MMSS 00, prop. 12.3)

Lemma

For every X𝕊 diaModX \in \mathbb{S}_{dia} Mod, the functor

X 𝕊 dia():𝕊 diaMod𝕊 diaMod X \wedge_{\mathbb{S}_{dia}}(-) \;\colon\; \mathbb{S}_{dia} Mod \longrightarrow \mathbb{S}_{dia} Mod

sends acylic cofibrations in the stable model structure to morphisms that are stable equivalences and h-cofibrations.

(MMSS 00, prop. 12.5)

Proposition

The symmetric monoidal smash product of spectra 𝕊 dia\wedge_{\mathbb{S}_{dia}} on 𝕊 diaMod\mathbb{S}_{dia} Mod, def. 3 satisfies the pushout-product axiom with respect to the stable model structure 𝕊 diaMod\mathbb{S}_{dia} Mod of theorem 1.

(MMSS 00, prop. 12.6)

Proof

That the pushout product of two stable cofibrations is again a stable cofibration is the content of lemma 16. It remains to show that if at least one of them is a stanble equivalence, def. 5, then so is the pushout-product. That follows with a laborious argument using the above lemmas (…).

(…)

17.-18.) Relation to Γ\Gamma-spaces

(…)

19.) Simplicial and topological diagram spectra

(…)

Part III. Symmetric monoidal categories and FSPs

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. 14 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 5.

  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. 11

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 functoriallity 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. 11)

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

then this takes the form

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

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 pointed topologically enriched category (def.), i.e. an enriched category over (Top cg */,,S 0)(Top_{cg}^{\ast/}, \wedge, S^0) from example 5. 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. 11.

  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 2:

    c,d𝒞)𝒞(c,d)F(c,d)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(c,d) \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 2:

    c𝒞F(c,c)equc𝒞F(c,c)AAAAAAAAc,dρ˜ (c,d)(d)c,dρ˜ d,c(c)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)}(d) }{\longrightarrow}} {\overset{\underset{c,d}{\sqcup} \tilde\rho_{d,c}(c)}{\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 𝒞\mathcal{C} be a 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. 12) 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/} 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)(d))c,dU(ρ˜ d,c(c))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)}(d)) }{\longrightarrow}} {\overset{\underset{c,d}{\sqcup} U(\tilde\rho_{d,c}(c))}{\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 3 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 pointed topologically enriched categories (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. 12) of example 3:

[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

and by taking the composition maps to be the morphisms induced by the 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

by observing that these equalize the two actions in the definition of the end.

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}.

First of all this yields a concise statement of the pointed topologically enriched Yoneda lemma (prop.)

Proposition

(topologically enriched Yoneda lemma)

Let 𝒞\mathcal{C} be a 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. 13, from the functor represented by cc to FF, and the value of FF on cc.

In terms of the ends (def. 12) 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. 11 is essentially dual to the proof of the next prop. 12.

Now that natural transformations are phrased in terms of ends (example 3), as is the Yoneda lemma (prop. 11), it is natural to consider the dual statement involvng coends:

Proposition

(co-Yoneda lemma)

Let 𝒞\mathcal{C} be a 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

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. 12 is componentwise the canonical action

𝒞(d,c)F(c)F(d) \mathcal{C}(d,c) \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,xF(c)), ( c \overset{}{\to} c_0,\; x \in 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 d=id c 0d = 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 \underset{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. 12 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. 11), 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. 12) 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

Because 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 also takes coends in the first argument and ends in the second argument, to ends (def. 12):

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 \,.
Proposition

(left Kan extension via coends)

Let 𝒞,𝒟\mathcal{C}, \mathcal{D} be 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. 12):

(Lan pF):cc𝒞𝒟(p(c),d)F(c). (Lan_p F) \;\colon\; c \;\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 3 and def. 13), then use the respect of Maps(,) *Maps(-,-)_\ast for ends/coends (remark 7), use the smash/mapping space adjunction (cor.), use the Fubini theorem (prop. 13) and finally use Yoneda reduction (prop. 11) 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} \,.

Monoidal topological categories

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 4 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. 11), 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:

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Definition

A (pointed) topological braided monoidal category, is a (pointed) topological monoidal category 𝒞\mathcal{C} (def. 14) 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:

(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. 15) 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. 16) it is called a closed monoidal category if for each X𝒳X\in \mathcal{X} the functor X()()XX \otimes(-)\simeq (-)\otimes X has a right adjoint, denoted

𝒞[X,]X()𝒞. \mathcal{C} \underoverset {\underset{[X,-]}{\longrightarrow}} {\overset{X\otimes(-)}{\longleftarrow}} {\bot} \mathcal{C} \,.

For any other object YY, the object [X,Y]𝒞[X,Y] \in \mathcal{C} is then called the internal hom object between XX and YY.

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. 16 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 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.

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. 16) 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 4, 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..

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 4.

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

  1. A monoid in (Ab, ,)(Ab, \otimes_{\mathbb{Z}}, \mathbb{Z}) (def. 19) is equivalently a ring.

  2. A commutative monoid in in (Ab, ,)(Ab, \otimes_{\mathbb{Z}}, \mathbb{Z}) (def. 19) is equivalently a commutative ring RR.

  3. An RR-module object in (Ab, ,)(Ab, \otimes_{\mathbb{Z}}, \mathbb{Z}) (def. 20) is equivalently an RR-module;

  4. The tensor product of RR-module objects (def. 21) is the standard tensor product of modules.

  5. The category of module objects RMod(Ab)R Mod(Ab) (def. 21) is the standard category of modules RModR Mod.

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 monoidal categories (def. 14). 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 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 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. 15), then the lax monoidal functor FF is called a braided monoidal functor if in addition the following 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 ) } \,.
Remark

In the literature often the term “monoidal functor” refers by default to what in def. 18 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. 16) then a braided monoidal functor (def. 18) between them is often called a symmetric monoidal functor.

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. 16) (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.

Definition

Given a (pointed) topological monoidal category (𝒞,,1)(\mathcal{C}, \otimes, 1), and given (A,μ,e)(A,\mu,e) a monoid in (𝒞,,1)(\mathcal{C}, \otimes, 1) (def. 19), 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 diagram commutes:

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

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

  2. (action property) the following 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 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.

Definition

Given a (pointed) topological symmetric monoidal category (𝒞,,1)(\mathcal{C}, \otimes, 1) (def. 16), given (A,μ,e)(A,\mu,e) a commutative monoid in (𝒞,,1)(\mathcal{C}, \otimes, 1) (def. 19), 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.19), then 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 1coequN 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{coequ}{\longrightarrow} N_1 \otimes_A N_2
Proposition

Given a (pointed) topological symmetric monoidal category (𝒞,,1)(\mathcal{C}, \otimes, 1) (def. 16), and given (A,μ,e)(A,\mu,e) a commutative monoid in (𝒞,,1)(\mathcal{C}, \otimes, 1) (def. 19). If all coequalizers exist in 𝒞\mathcal{C}, then the tensor product of modules A\otimes_A from def. 21 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.

Definition

Given a monoidal category of modules (AMod, A,A)(A Mod , \otimes_A , A) as in prop. 15, then a monoid (E,μ,e)(E, \mu, e) in (AMod, A,A)(A Mod , \otimes_A , A) (def. 19) is called an AA-algebra.

Propposition

Given a 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. 15, and an AA-algebra (E,μ,e)(E,\mu,e) (def. 22), 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 }

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 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.

Day convolution

Definition

Let 𝒞\mathcal{C} be a small pointed topological monoidal category (def. 14) with tensor product denoted 𝒞:𝒞×𝒞𝒞\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 [𝒞,Top cg */][\mathcal{C},Top^{\ast/}_{cg}] (def. 13) 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. 11) given by the following coend (def. 12)

X DayY:c(c 1,c 2)𝒞×𝒞𝒞(c 1 𝒞c 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_{\mathcal{C}} 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:

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. 23 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. 14). This will be key for understanding monoids and modules with respect to Day convolution.

Definition

Let 𝒞\mathcal{C} be a 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}]

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

The Day convolution product (def. 23) of two functors is equivalently the left Kan extension (def. 14) of their external tensor product (def. 24) along the tensor product 𝒞\otimes_{\mathcal{C}}: there is a natural isomorphism

X DayYLan 𝒞(X¯Y). X \otimes_{Day} Y \simeq Lan_{\otimes_{\mathcal{C}}} (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. 14 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)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) \wedge X(c_1)\wedge X(c_2) \end{aligned} \,.
Corollary

The Day convolution Day\otimes_{Day} (def. 23) 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 \wedge ) \,,

where ¯\overline{\wedge} is the external product of def. 24.

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 corepresented functor y(c):d𝒞(c,d)y(c)\colon d \mapsto \mathcal{C}(c,d).

Proposition

For 𝒞\mathcal{C} a small pointed topological monoidal category (def. 14), the Day convolution tensor product Day\otimes_{Day} of def. 23 makes the pointed topologically enriched functor category

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

a pointed topological monoidal category (def. 14) with tensor unit y(1)y(1) co-represented by the tensor unit 11 of 𝒞\mathcal{C}.

Proof

Regarding associativity, observe that

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

where we used the Fubini theorem for coends (prop. 13) and then twice the co-Yoneda lemma (prop. 12). An analogous formula follows for X Day(Y DayZ)))(c)X \otimes_{Day} (Y \otimes_{Day} Z)))(c), and so associativity follows via prop. 17 from the associativity of the smash product and of the tensor product 𝒞\otimes_{\mathcal{C}}.

To see that y(1)y(1) is the tensor unit for Day\otimes_{Day}, use the Fubini theorem for coends (prop. 13) and then twice the co-Yoneda lemma (prop. 12) 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 1𝒞X(c 1)c 2𝒞𝒞(c 1 𝒞c 2,)𝒞(1,c 2) c 1𝒞X(c 1)𝒞(c 1 𝒞1,) c 1𝒞X(c 1)𝒞(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} X(c_1) \wedge \overset{c_2 \in \mathcal{C}}{\int} \mathcal{C}(c_1\otimes_{\mathcal{C}} c_2,-) \wedge \mathcal{C}(1,c_2) \\ & \simeq \overset{c_1\in \mathcal{C}}{\int} X(c_1) \wedge \mathcal{C}(c_1 \otimes_{\mathcal{C}} 1, -) \\ & \simeq \overset{c_1\in \mathcal{C}}{\int} X(c_1) \wedge \mathcal{C}(c_1, -) \\ & \simeq X(-) \\ & \simeq X \end{aligned} \,.
Proposition

For 𝒞\mathcal{C} a small pointed topological monoidal category (def. 14) with tensor product denoted 𝒞:𝒞×𝒞𝒞\otimes_{\mathcal{C}} \;\colon\; \mathcal{C} \times\mathcal{C} \to \mathcal{C}, the monoidal category with Day convolution ([𝒞,Top cg */], Day,y(1))([\mathcal{C},Top^{\ast/}_{cg}], \otimes_{Day}, y(1)) from def. 18 is a closed monoidal category (def. 17). Its internal hom [,] Day[-,-]_{Day} is given by the end (def. 12)

[X,Y] Day(c)c 1,c 2Maps(𝒞(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_{\mathcal{C}} c_1,c_2), \; Maps(X(c_1) , Y(c_2))_\ast \right)_\ast \,.
Proof

Using the Fubini theorem (def. 13) and the co-Yoneda lemma (def. 12) and in view of definition 13 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(c 𝒞c 1,c 2),Maps(Y(c 1),Z(c 2)) *) *) * cc 1,c 2Maps(𝒞(c 𝒞c 1,c 2)X(c)Y(c 1),Z(c 2)) * c 2Maps(c,c 1𝒞(c 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_{\mathcal{C}} 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_{\mathcal{C}} 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_{\mathcal{C}} 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. 18, the Yoneda embedding c𝒞(c,)c\mapsto \mathcal{C}(c,-) constitutes a strong monoidal functor (def. 18)

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

That the tensor unit is respected is part of prop. 18. To see that the tensor product is respected, apply the co-Yoneda lemma (prop 12) twice to get the following natural isomorphism

(y(c 1) Dayy(c 2))(c) d 1,d 2𝒞(d 1 𝒞d 2,c)𝒞(c 1,d 1)𝒞(c 2,d 2) 𝒞(c 1 𝒞c 2,c) =y(c 1 𝒞c 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_{\mathcal{C}} d_2, c ) \wedge \mathcal{C}(c_1,d_1) \wedge \mathcal{C}(c_2,d_2) \\ & \simeq \mathcal{C}(c_1\otimes_{\mathcal{C}}c_2 , c ) \\ & = y(c_1 \otimes_{\mathcal{C}} c_2 )(c) \end{aligned} \,.

Functors with smash product

Proposition

Let (𝒞,I)(\mathcal{C},\otimes I) be a pointed topologically enriched category (symmetric monoidal category) monoidal category (def. 14). Regard (Top cg */,,S 0)(Top_{cg}^{\ast/}, \wedge , S^0) as a topological symmetric monoidal category as in example 5.

Then (commutative) monoids in (def. 19) 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. 18 are equivalent to (braided) lax monoidal functors (def. 18) of the form

(𝒞,,I)(Top cg *,,S 0), (\mathcal{C},\otimes, I) \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.

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 18, 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 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. 11) 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 3 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. 18 satisfied by a lax monoidal functor through these identifications gives precisely the conditions of def. 19 on a (commutative) 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. 18) between pointed topologically enriched monoidal categories (def. 14). 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 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. 16) and ff is a braided monoidal functor (def. 18), then f *f^\ast also preserves 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}}) \,.
Proof

This is an immediate corollary of prop. 21, since the composite of two (braided) lax monoidal functors is itself canonically a (braided) lax monoidal functor.

Monoidal homotopy theory

We now combine the concepts of model category and monoidal category.

Definition

A (symmetric) monoidal model category is model category 𝒞\mathcal{C} equipped with the structure of a closed symmetric monoidal category (𝒞,,I)(\mathcal{C}, \otimes, I) 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

    (XY) XX(YX)YY, (X \otimes Y') \coprod_{X \otimes X'} (Y \otimes X') \longrightarrow Y \otimes Y' \,,

    is itself a cofibration, which, furthermore, is acyclic if ff or ff' is.

    (Equivalently this says that the tensor product :C×CC\otimes : C \times C \to C is a left Quillen bifunctor.)

  2. (unit axiom) For every cofibrant object XX and every cofibrant resolution QIp I*\emptyset \hookrightarrow Q I \stackrel{p_I}{\longrightarrow} \ast of the tensor unit II, the resulting morphism

    QIXp IXIXX Q I \otimes X \stackrel{p_I \otimes X}{\longrightarrow} I\otimes X \stackrel{\simeq}{\longrightarrow} X

is a weak equivalence.

Remark

The pushout-product axiom in def. 25 implies that for XX a cofibrant object, then the functor X()X \otimes (-) preserves cofibrations and acyclic cofibrations.

In particular if the tensor unit II happens to be cofibrant, then the unit axiom in def. 25 is implied by the pushout-product axiom.

Definition

We say a monoidal model category, def. 25, satisfies the monoid axiom, def. 25, if every morphism that is obtained as a transfinite composition of pushouts of tensor products XfX\otimes f of acyclic cofibrations ff with any object XX is a weak equivalence.

(Schwede-Shipley 00, def. 3.3.).

Remark

In particular, the axiom in def. 26 says that for every object XX the functor X()X \otimes (-) sends acyclic cofibrations to weak equivalences.

Proposition

Let (𝒞,,I)(\mathcal{C}, \otimes, I) be a monoidal model category. Then the left derived functor of the tensor product exsists and makes the homotopy category into a monoidal category (Ho(𝒞), L,γ(I))(Ho(\mathcal{C}), \otimes^L, \gamma(I)).

If in in addition (𝒞,)(\mathcal{C}, \otimes) satisfies the monoid axiom, then the localization functor γ:𝒞Ho(𝒞)\gamma\colon \mathcal{C}\to Ho(\mathcal{C}) carries the structure of a lax monoidal functor

γ:(𝒞,,I)(Ho(𝒞), L,γ(I)). \gamma \;\colon\; (\mathcal{C}, \otimes, I) \longrightarrow (Ho(\mathcal{C}), \otimes^L , \gamma(I)) \,.
Proof

Consider the explicit model of Ho(𝒞)Ho(\mathcal{C}) as the category of fibrant-cofibrant objects in 𝒞\mathcal{C} with left/right-homotopy classes of morphisms between them (discussed at homotopy category of a model category).

A derived functor exists if its restriction to this subcategory preserves weak equivalences. Now the pushout-product axiom implies that on the subcategory of cofibrant objects the functor \otimes preserves acyclic cofibrations, and then the preservation of all weak equivalences follows by Ken Brown's lemma.

Hence L\otimes^L exists and its associativity follows simply by restriction. It remains to see its unitality.

To that end, consider the construction of the localization functor γ\gamma via a fixed but arbitrary choice of (co-)fibrant replacements QQ and RR, assumed to be the identity on (co-)fibrant objects. We fix notation as follows:

Cofi XQXWFibp xX,XWCofj XRXFibq x*. \emptyset \underoverset{\in Cof}{i_X}{\longrightarrow} Q X \underoverset{\in W \cap Fib}{p_x}{\longrightarrow} X \;\;\,,\;\; X \underoverset{\in W \cap Cof}{j_X}{\longrightarrow} R X \underoverset{\in Fib}{q_x}{\longrightarrow} \ast \,.

Now to see that γ(I)\gamma(I) is the tensor unit for L\otimes^L, notice that in the zig-zag

(RQI)(RQX)j QI(RQX)(QI)(RQX)(QI)j QX(QI)(QX)p I(QX)IQXQX (R Q I) \otimes (R Q X) \overset{j_{Q I} \otimes (R Q X)}{\longleftarrow} (Q I) \otimes (R Q X) \overset{(Q I)\otimes j_{Q X}}{\longleftarrow} (Q I) \otimes (Q X) \overset{p_I \otimes (Q X)}{\longrightarrow} I \otimes Q X \simeq Q X

all morphisms are weak equivalences: For the first two this is due to the pushout-product axiom, for the third this is due to the unit axiom on a monoidal model category. It follows that under γ()\gamma(-) this zig-zig gives an isomorphism

γ(I) Lγ(X)γ(X) \gamma(I) \otimes^L \gamma(X)\simeq \gamma(X)

and similarly for tensoring with γ(I)\gamma(I) from the right.

To exhibit lax monoidal structure on γ\gamma, we need to construct a natural transformation

γ(X) Lγ(Y)γ(XY) \gamma(X) \otimes^L \gamma(Y) \longrightarrow \gamma(X \otimes Y)

and show that it satisfies the the appropriate associativity and unitality condition.

By the definitions at homotopy category of a model category, the morphism in question is to be of the form

(RQX)(RQY)RQ(XY) (R Q X) \otimes (R Q Y) \longrightarrow R Q (X\otimes Y)

To this end, consider the zig-zag

(RQX)(RQY)CofWj QXRQY(QX)(RQY)CofW(QX)j QY(QX)(QY)p X(QY)X(QY)Yp YXY, (R Q X) \otimes (R Q Y) \underoverset{\in Cof \cap W}{j_{Q X} \otimes R Q Y}{\longleftarrow} (Q X) \otimes (R Q Y) \underoverset{\in Cof \cap W}{(Q X) \otimes j_{Q Y} }{\longleftarrow} (Q X) \otimes (Q Y) \overset{p_X \otimes (Q Y)}{\longrightarrow} X \otimes (Q Y) \overset{Y \otimes p_Y}{\longrightarrow} X \otimes Y \,,

and observe that the two morphisms on the left are weak equivalences, as indicated, by the pushout-product axiom satisfied by \otimes.

Hence applying γ\gamma to this zig-zag, which is given by the two horizontal part of the following digram

(RQX)(RQY) R(QXQY) RQ(XY) id j QXQY j Q(XY) id id p XY (RQX)(RQY) CofWj QXj QY (QX)(QY) p Xp Y XY, \array{ (R Q X) \otimes (R Q Y) &\longleftarrow& R( Q X \otimes Q Y ) &\longrightarrow& R Q (X \otimes Y) \\ \uparrow^{\mathrlap{id}} && \uparrow^{\mathrlap{j_{Q X \otimes Q Y}}} && \uparrow^{\mathrlap{j_{Q(X \otimes Y)}}} \\ \downarrow^{\mathrlap{id}} && \downarrow^{\mathrlap{id}} && \downarrow^{\mathrlap{p_{X\otimes Y}}} \\ (R Q X) \otimes (R Q Y) &\underoverset{\in Cof \cap W}{j_{Q X} \otimes j_{Q Y}}{\longleftarrow}& (Q X) \otimes (Q Y) &\overset{p_X \otimes p_Y}{\longrightarrow}& X \otimes Y } \,,

and inverting the first two morphisms, this yields a natural transformation as required.

To see that this satisfies associativity if the monoid axiom holds, tensor the entire diagram above on the right with (RQZ)(R Q Z) and consider the following pasting composite:

(RQX)(RQY)(RQZ) R(QXQY)(RQZ) (RQ(XY))(RQZ) id j QXQYid j Q(XY)id Q(XY)(RQZ) idj QZ Q(XY)(QZ) id id p (XY)id () p (XY)id (RQX)(RQY)(RQZ) CofWj QXj QYid (QX)(QY)(RQZ) p Xp Yid XY(RQZ) idj QZ XYQZ idp Z XYZ, \array{ (R Q X) \otimes (R Q Y) \otimes (R Q Z) &\longleftarrow& R( Q X \otimes Q Y ) \otimes (R Q Z) &\longrightarrow& (R Q (X \otimes Y)) \otimes (R Q Z) \\ \uparrow^{\mathrlap{id}} && \uparrow^{\mathrlap{j_{Q X \otimes Q Y} \otimes id }} && \uparrow^{\mathrlap{j_{Q(X \otimes Y)}\otimes id }} \\ && && Q(X \otimes Y) \otimes (R Q Z) &\overset{id \otimes j_{Q Z}}{\longleftarrow}& Q(X\otimes Y) \otimes (Q Z) \\ \downarrow^{\mathrlap{id}} && \downarrow^{\mathrlap{id}} && \downarrow^{\mathrlap{p_{(X\otimes Y)} \otimes id }} &(\star)& \downarrow^{\mathrlap{p_{(X \otimes Y)} \otimes id}} \\ (R Q X) \otimes (R Q Y) \otimes (R Q Z) &\underoverset{\in Cof \cap W}{j_{Q X} \otimes j_{Q Y} \otimes id}{\longleftarrow}& (Q X) \otimes (Q Y) \otimes (R Q Z) &\overset{p_X \otimes p_Y \otimes id}{\longrightarrow}& X \otimes Y \otimes (R Q Z) &\underset{id \otimes j_{Q Z}}{\longleftarrow}& X\otimes Y \otimes Q Z &\overset{id \otimes p_Z}{\longrightarrow}& X \otimes Y \otimes Z } \,,

Observe that under γ\gamma the total top zig-zag in this diagram gives

(γ(X) Lγ(Y)) Lγ(Z)γ(XY) Lγ(Z). (\gamma(X) \otimes^L \gamma(Y)) \otimes^L \gamma(Z) \to \gamma(X\otimes Y)\otimes^L \gamma(Z) \,.

Now by the monoid axiom (but not by the pushout-product axiom!), the horizontal maps in the square in the bottom right (labeled \star) are weak equivalences. This implies that the total horizontal part of the diagram is a zig-zag in the first place, and that under γ\gamma the total top zig-zag is equal to the image of that total bottom zig-zag. But by functoriality of \otimes, that image of the bottom zig-zag is

γ(p Xp Yp Z)γ(j QXj QYj QZ) 1. \gamma(p_X \otimes p_Y \otimes p_Z) \circ \gamma(j_{Q X} \otimes j_{Q Y} \otimes j_{Q Z})^{-1} \,.

The same argument applies to left tensoring with RQZR Q Z instead of right tensoring, and so in both cases we reduce to the same morphism in the homotopy category, thus showing the associativity condition on the transformation that exhibits γ\gamma as a lax monoidal functor.

category: reference

Revised on July 28, 2016 06:53:54 by Urs Schreiber (131.220.184.222)