on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
on algebras over an operad, on modules over an algebra over an operad
on dendroidal sets, for dendroidal complete Segal spaces, for dendroidal Cartesian fibrations
This page collects material related to
Michael Mandell, Peter May, Stefan Schwede, Brooke Shipley,
Model categories of diagram spectra,
Proceedings of the London Mathematical Society Volume 82, Issue 2, 2000
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
Robert Piacenza, Homotopy theory of diagrams and CW-complexes over a category, Can. J. Math. Vol 43 (4), 1991 (pdf)
also chapter VI of Peter May et al., Equivariant homotopy and cohomology theory, 1996 (pdf)
Lydakis, Simplicial functors and stable homotopy theory Preprint, available via Hopf archive, 1998 (pdf)
Mark Hovey, Brooke Shipley, Jeff Smith, Symmetric spectra, J. Amer. Math. Soc. 13 (2000), 149-208 (arXiv:math/9801077)
Stefan Schwede, Symmetric spectra (2012)
This part gives a unified discussion of the categories of
(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 $Top^{\ast/}_{fin}$.
Throughout, write
$Top \coloneqq Top_{cg}$ for the category of compactly generated topological spaces; this is a cartesian monoidal category $(Top, \times)$ (in fact a cartesian closed category);
$Top^{\ast/}$ for the corresponding pointed topological spaces; this is a symmetric monoidal category equipped with the smash product $\wedge$ of pointed objects;
$Top_{Quillen}$ for the classical model structure on topological spaces (compactly generated topological spaces); in particular this is a monoidal model category $(Top_{Quillen}, \times)$;
$Top^{\ast/}_{Quillen}$ for the corresponding classical model structure on pointed topological spaces, in particular this is a monoidal model category $(Top_{Quillen}^{\ast/}, \wedge)$.
Throughout part I we are dealing with $(Top^{\ast/},\wedge)$-enriched categories, $(Top^{\ast/}, \wedge)$-enriched functors, etc., and then in part II we are dealing with $(Top_{Quillen}^{\ast/}, \wedge)$-enriched model categories etc.
Define the following pointed topologically enriched symmetric closed monoidal categories (the tensor product is a pointed topologically enriched functor):
$Seq$ 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 $Top^{\ast/}$-enriched category the hom-spaces are
$Sym$ is the standard skeleton of the core of FinSet, objects are the sets $\{1, \cdots,n\}$ for $n \in \mathbb{N}$, all morphisms are automorphisms and the automorphism group of $\{1,\cdots,n\}$ is the symmetric group $\Sigma_n$, tensor product is the disjoint union of sets, tensor unit is the empty set; we turn this into a $Top^{\ast/}$-enriched category by adjoining a basepoint:
$Orth$ has as objects finite dimenional real linear inner product spaces $(V, \langle -,-\rangle)$ and as morphisms the linear isometric isomorphisms between these; hence the automorphism group of the object $(V, \langle -,-\rangle)$ is the orthogonal group $O(V)$; the monoidal product is direct sum of linear spaces, the tensor unit is the 0-vector space; again we turn this into a $Top^{\ast/}$-enriched category by adjoining a basepoint to the hom-spaces;
Denote the canonical faithful subcategory inclusions by
where $S^V$ denotes the one-point compactification of $V$. On morphisms $sym \colon (\Sigma_n)_+ \hookrightarrow (O(n))_+$ is the inclusion of permutation matrices into orthogonal matrices and $orth \colon O(V)_+ \hookrightarrow Aut(S^V)$ is on $O(V)$ the topological subspace inclusions of the pointed homeomorphisms $S^V \to S^V$ that are induced under forming one-point compactification from linear isometries of $V$.
The sequence of inclusions in def. 1 satisfies the following properties:
All three inclusions are strong monoidal functors.
Under passing to enriched functor categories, restriction $(-)^\ast$ along these inclusions and left Kan extension $(-)_!$ along them yields a sequence of adjunctions
All four enriched functor categories become symmetric monoidal categories with the Day convolution monoidal product structure induced by the monoidal structure of their sites.
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:
For $\mathcal{C}$ a $V$-enriched monoidal category, under the Yoneda embedding
the tensor unit in $\mathcal{C}$ goes to the tensor unit of the induced Day convolution structure on $[\mathcal{C}, Top^{\ast/}]$.
(see at Day convolution this lemma)
Write
for the image under the Yoneda embedding of the tensor unit in $Top_{fin}^{\ast/}$ with its smash product (the 0-sphere), which by lemma 1 is the tensor unit in $([Top_{fin}^{\ast/}, Top^{\ast/}], \otimes_{Day})$.
Since this is going to be the standard presentation of the sphere spectrum in the model structure for excisive functors on $[Top_{fin}^{\ast/}, Top^{\ast/}]$ we refer to it as the sphere spectrum.
For its restrictions along the above sub-site inclusions, prop. 1, write
While $\mathbb{S}$ in def. 2 is the tensor unit in $([Top_{fin}^{\ast/}, Top^{\ast/}], \otimes_{Day})$, neither of its restrictions $\mathbb{S}_{Orth},\mathbb{S}_{Sym}, \mathbb{S}_{Seq}$ is the tensor unit in $([Orth, Top^{\ast/}],\otimes_{Day}), ([Sym, Top^{\ast/}],\otimes_{Day}), ([Seq, Top^{\ast/}],\otimes_{Day})$, respectively.
Nevertheless, because by prop. 1 the restriction functors are strong monoidal functors and because the tensor unit $\mathbb{S}$ canonically has the structure of a monoid object, each of $\mathbb{S}_{Orth},\mathbb{S}_{Sym}, \mathbb{S}_{Seq}$ inherts the structure of a monoid object in the respective Day convolution monoidal category.
Moreover, $\mathbb{S}_{Orth}$ and $\mathbb{S}_{Sym}$ are commutative monoid objects, while $\mathbb{S}_{Seq}$ 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 $Seq$).
$\mathbb{S}$ | $\mathbb{S}_{Orth}$ | $\mathbb{S}_{Sym}$ | $\mathbb{S}_{Seq}$ | |
---|---|---|---|---|
monoid object | yes | yes | yes | yes |
commutative monoid object | yes | yes | yes | no |
tensor unit | yes | no | no | no |
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 $\mathbb{S}_{Orth}$ is the one given by the canonical natural transformations
and $\mathbb{S}_{Sym}$ and $\mathbb{S}_{Seq}$ the ones given by the canonical natural transformations
Therefore we may consider module objects over the restrictions of the sphere spectrum from def. 2.
The category of right module objects over $\mathbb{S}_{Orth}$, $\mathbb{S}_{Sym}$ and $\mathbb{S}_{Seq}$ 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):
Write $\mathbb{S}_{dia}$ 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 $\mathbb{S}_{dia}$ as in remark 1. This means that for $\mathbb{S}_{Sym}$ and $\mathbb{S}_{Seq}$ they are functors $X \colon Sym \longrightarrow sSet^{\ast/}$ or $X \colon Seq \longrightarrow sSet^{\ast/}$, respectively equipped with natural transformations
satisfying the evident categorified action property. In the present case this action property says that these morphisms are determined by
under the isomorphisms $S^p \simeq S^1 \wedge S^{p-1}$. Naturality of all these morphisms as functors on $Sym$ is the equivariance under the symmetric group actions in the definition of symmetric spectra.
Similarly, modules over $\mathbb{S}_{Orth}$ are equivalently functors
etc. and their functoriality embodies the orthogonal group-equivariance in the definition of orthogonal spectra.
For completeness, we may, trivially, add to the three statements in prop. 2 the equivalence
which holds tautologically because by def. 2 $\mathbb{S}$ is in fact the tensor unit in $([Top^{\ast/}_{fin}, Top^{\ast/}],\otimes_{Day})$, so that every object here is canonically a module object over $\mathbb{S}$.
Now the model structure on excisive functors shows that the category $[Top^{\ast/}_{fin}, Top^{\ast/}]$ 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.
By remark 1 the categories $\mathbb{S}_{Sym} Mod_r$, $\mathbb{S}_{Orth} Mod_r$ and $\mathbb{S}_{Orth} Mod_r$ are categories of modules over a commutative monoid object and as such they inherit symmetric monoidal category structure
Via prop. 2 this is equivalently symmetric monoidal product structure
on the category of symmetric spectra and
on that of orthogonal spectra. This is called the symmetric monoidal smash product of spectra.
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
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.
We write $dia \in \{Top^{\ast/, Orth, Sym, Seq}\}$ generically for any one of the four sites in def. 1.
Accordingly we write $\mathbb{S}_{dia} \in \{\mathbb{S}, \mathbb{S}_{Orth}, \mathbb{S}_{Sym}, \mathbb{S}_{Seq}\}$ generically for any one of the four incarnations of the sphere spectrum according to def. 2, over these sites.
Finally we will write
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:
Let ${dia} \in \{Top^{\ast/}, {Orth}, {Sym}, {Seq}\}$ be any one of the sites in def. 1.
There is an equivalence of categories
between the corresponding category of modules (prop. 2) and the $Top^{\ast/}$-enriched functor category over the opposite of its full subcategory on the free modules.
If $\mathbb{S}_{dia}$ is a commutative monoid (hence for $dia \in \{Top^\ast/, Orth,Sym\}$ but not for $dia = Seq$) then $\mathbb{S}_{dia} FreeMod^{op}$ carries a symmetric monoidal category structure such that under the above equivalence its Day convolution is the tensor product of modules.
There is a $Top^{\ast/}$-enriched functor
which is monoidal when $\mathbb{S}_{dia}$ is commutative and which is such that the (free module$\dashv$forgetful functor)-adjunction is equivalently the base change along $\delta$:
This is a general statement about modules with respect to Day convolution monoidal structures, see this proposition (MMSS 00, theorem 2.2).
For the sequential case $dia = Seq$, then the opposite category $\mathbb{S}_{Seq} FreeMod^{op}$ of free modules over $\mathbb{S}_{Seq}$ (def. 2) in lemma 2 is identified as the non-full subcategory $StdSpheres$ of $Top^{\ast/}$ whose objects are the standard spheres $S^n \coloneqq (S^1)^{\wedge^n}$ and whose hom spaces are the canonical image of $S^{n_2-n_1}$ in the hom space $Top^{\ast/}(S^{n_1},S^{n_2})$ (the image under the smash$\dashv$hom-adjunct of the identity) if $n_2 \gt n_1$, and the point otherwise:
Hence according to lemma 2 sequential spectra are equivalently $Top^{\ast/}$-enriched functors on StdSpheres:
In this form the statement appears also as (Lydakis 98, prop. 4.3). See also at sequential spectra – As diagram spectra.
The free $\mathbb{S}_{Dia}$ modules are those of the form $y(d)\wedge \mathbb{S}_{Dia}$, where $d \in Dia$ is an object, and $y$ denotes the Yoneda embedding. For the case $Dia = Seq$ this means that they are labeled by $k \in \mathbb{N}$, their underlying functor $F(k) \in [Seq,Top^{\ast/}]$ is given by
and the $\mathbb{S}_{Seq}$-action on these is, expressed as modules over monoidal functors (via this fact about modules with respect to Day convolution) by the canonical identifications
whenever $n_2 \geq k$, and by the zero morphism otherwise.
Now for $R$ any other $\mathbb{S}_{Seq}$-module, with action
then a homomorphism of $\mathbb{S}_{Seq}$-modules $\phi \colon F(k)\to R$ has components $\phi_{n}\colon S^{n-k} \to R_n$ fitting into commuting diagrams of the form
for $n_2 \geq k$. By the fact that the top morphism here is an isomorphism, all the components $\phi_{n \geq k}$ are uniquely fixed by the component $\phi_k$ as
(and $\phi_{n \lt k} = 0$). Of course this just confirms the free property of free spectra: morphisms of $\mathbb{S}_{Seq}$-modules $F(k) \longrightarrow R$ are equivalent to morphisms in $[Seq,Top^{\ast/}]$ from $y(k)$ to $R$, which by the Yoneda lemma form the space $R_k \in Top^{\ast/}$.
Specialized to $R$ a free spectrum itself this verifies that the hom-spaces between free $\mathbb{S}_{Seq}$-modules are as claimed:
Of course this is just the defining free property. But now comparison with the above commuting square for the case $n_1 = k_1$ and $n_2 = k_2$ shows that composition in $\mathbb{S}_{Seq} FreeMod^{op}$ is indeed the composition in $Top^{\ast/}$ under the identification of $F(k)$ with $S^k$ and of the above hom-space $S^{k_2-k_1}$ with its image under $S^{k_2-k_1} \to Top^{\ast/}(S^{k_1},S^{k_2})$.
From lemma 2 we immediately get the following:
The horizontal adjunctions in remark 3 lift to adjunctions between categories of modules, such as to give a commuting diagram of adjunctions as follows:
(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.
Write $[Seq, Top^{\ast/}_{Quillen}]$ 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 $[Seq, Top^{\ast/}_{Quillen}]$ along the composite adjunction connecting the two in that diagram, so that we get a diagram of Quillen adjunctions
Equivalently, under the identification
of lemma 2, the strict model structure is the projective model structure on enriched functors (Piacenza 91) see hereodel structure on topological spaces#ModelStructureOnTopEnrichedFunctors).
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.
The model structure
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.
For any of the four categories of spectra in prop. 3, we say (with notation from def. 4) that:
an object $X$ is an Omega-spectrum if $seq^\ast X$ is an Omega spectrum in the standard sense of sequential spectra;
a morphism $f$ is a stable weak homotopy equivalence if $seq^\ast(f)$ is a stable weak homotopy equivalence in the standard sense of sequential spectra (an isomorphism on all stable homotopy groups of sequential spectra);
a morphism $f$ is a stable equivalence if for all Omega-spectra $X$ in the above sense, the morphism $[f,E]_{strict}$ is a bijection (where $[-,E]_{strict}$ 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.
The weak equivalences in the strict model structure (prop. 4) and the stable equivalences of def. 5 are related as follows:
Every weak equivalence with respect to the strict model structure is a stable equivalence.
Every stable equivalence between Omega-spectra (def. 5) is a weak equivalence in the strict model structure.
The first statement follows by def. 5 since weak equivalences become isomorphisms in the homotopy category.
For the second statement, let $f \colon X \to Y$ be a stable equivalence between Omega-spectra. Then by definition, in particular
is a bijection. Therefore the pre-image of $[id_X] \in [X,X]_{strict}$ is an inverse to $f$ in the homotopy category of the strict model structure. Hence $f$ represents an isomorphism in the strict homotopy category and is hence a weak equivalence in the strict model structure.
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).
For $dia \in \{Top^{\ast/}_{fin}, Orth, Sym, Seq\}$ and for each $n \in \mathbb{N}$, the functor
that sends a structured spectrum (notation as in def. 4) to the $n$th component space of its underlying sequential spectrum has a left adjoint
This is called the free structured spectrum-functor.
The adjunction units in the sequence of adjunctions equip these with canonical natural transformations
Under the equivalence of lemma 2
the free spectrum on $K \in Top^{\ast/}$ is
where $y(n)$ denotes the functor represented by $dia(n)$ (i.e. by $n$ if $dia = Seq$, by $\{1,\cdots,n\}$ if $dia = Sym$, by $\mathbb{R}^n$ if $dia = Orth$ and by $S^n$ if $dia = Top^{\ast/}_{fin}$).
On the free $\mathbb{S}_{dia}$-module on $e \in Dia^{op} \stackrel{y}{\hookrightarrow} [Dia, Top^{\ast/}]$ this takes the value
(MMSS00, p. 7 with theorem 2.2)
Generally, for $\mathcal{C}$ a $V$-enriched symmetric monoidal category, and for $c\in \mathcal{C}$ an object, then there is an adjunction
whose characterizing natural isomorphism is the combination of that of tensoring with the Yoneda embedding:
Specializing this to our case with $V = Top^{\ast/}$ and $\mathcal{C} = \mathbb{S}_{dia} FreeMod^{op}$ yields the first statement. The second follows from similar yoga, see the formula at Day convolution – Modules.
Explicitly, the free spectra according to def. 6, look as follows:
For sequential spectra: $(F^{Seq}_n K)_q \simeq K \wedge S^{q-n}$;
for symmetric spectra: $(F^{Sym}_n K)_q \simeq \Sigma(q)_+ \wedge_{\Sigma(q-n)} K \wedge S^{q-n}$.
for orthogonal spectra: $(F^{Orth}_n K)_q \simeq O(q)_+ \wedge_{O(q-n)} K \wedge S^{q-n}$.
In particular:
$F_0 K = \Sigma^\infty K$;
$F_n^{Seq}S^n$ is like the suspension spectrum of the point (the standard sequential sphere spectrum) but with its first $n$ components simply removed.
(e.g. Schwede 12, example 3.20)
By working out the formula in item 2 of lemma 3. In the sequential case $(Dia = Seq)$ there exists a morphism $k_1 \otimes k_2 \to k_3$ only if $k_1 + k_2 = k_3$ and then there is a unique such. Hence here the colimit in the formula becomes a coproduct and we find
In the symmetric case ($Dia = Sym$) the formula is similar except that $S^0 = Seq(q,q)_+$ is replaced by $Sym(q,q)_+ = \Sigma(q)_+$ and the colimit goes over the automorphisms that fix $q-n$ elements, thereby producing the partial smash tensor shown in the statement. Analogously for the orthogonal case ($Dia = Orth$).
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.
For each $n \in \mathbb{N}$ there exists a morphism
such that for every $X\in \mathbb{S}_{dia} Mod$ the operation $\lambda_n^\ast$ of precomposition with $\lambda_n$ forms a commuting diagram of the form
where the horizontal equivalences are the adjunction isomorphisms and the canonical identification, and where the right morphism is the $(\Sigma \dashv \Omega)$-adjunct of the structure map $\sigma_n$ of the sequential spectrum $seq^\ast X$ underlying $X$ (def. 4).
Since all prescribed morphisms in the diagram are natural transformations, this is in fact a diagram of copreheaves on $\mathbb{S}_{dia} Mod$
With this the statement follows by the Yoneda lemma.
Now we say explicitly what these maps are:
For $n \in \mathbb{N}$, write
for the adjunct under the (free structured spectrum $\dashv$ $n$-component)-adjunction in def. 6 of the composite morphism
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)
(MMSS 00, lemma 8.5, following Hovey-Shipley-Smith 00, remark 2.2.12)
Consider the case $Dia = Seq$ and $n = 0$. All other cases work analogously.
By prop. 6, in this case the morphism $\lambda_0$ has components like so:
Now for $X$ any sequential spectrum, then a morphism $f \colon F_0 S^0 \to X$ is uniquely determined by its 0th component $f_0 \colon S^0 \to X_0$ (that’s of course the free property of $F_0 S^0$); as the compatibility with the structure maps forces the first component, in particular, to be $\sigma_0^X\circ \Sigma f$:
But that first component is just the component that similarly determines the precompositon of $f$ with $\lambda_0$, hence $\lambda_0^\ast f$ is fully fixed as being the map $\sigma_0^X \circ \Sigma f$. Therefore $\lambda_0^\ast$ is the function
It remains to see that this is the $(\Sigma \dashv \Omega)$-adjunct of $\sigma_0^X$. By the general formula for adjuncts, this is
To compare to the above, we check what this does on points: $S^0 \stackrel{f_0}{\longrightarrow} X_0$ is sent to the composite
To identify this as a map $S^1 \to X_1$ we use the adjunction isomorphism once more to throw all the $\Omega$-s on the right back to $\Sigma$-s the left, to finally find that this is indeed
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.
The maps $\lambda_n \;\colon\; F_{n+1} S^1 \longrightarrow F_n S^0$ in def. 7 are
stable equivalences, according to def. 5, for all four cases of spectra, ${Dia} \in \{Top^{\ast/}, Orth, Sym, Seq\}$;
stable weak homotopy equivalences, according to def. 5, for sequential spectra, symmetric spectra and pre-excisive functors ${Dia} \in \{Top^{\ast/}, Orth, Seq\}$;
not stable weak homotopy equivalences for the case of symmetric spectra ${Dia} = {Sym}$.
(Hovey-Shipley-Smith 00, example 3.1.10, MMSS 00, lemma 8.6, Schwede 12, example 4.26)
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
and this immediately gives that $\lambda_n$ is an isomorphism on stable homotopy groups.
orthogonal case
Here for $q \geq n+1$ the $q$-component of $\lambda_n$ is the quotient map
By the suspension isomorphism for stable homotopy groups, $\lambda_n$ is a stable weak homotopy equivalence precisely if any of its suspensions is. Hence consider instead $\Sigma^n \lambda_n \coloneqq S^n \wedge \lambda_n$, whose $q$-component is
Now due to the fact that $O(q-k)$-action on $S^q$ lifts to an $O(q)$-action, the quotients of the diagonal action of $O(q-k)$ equivalently become quotients of just the left action. Formally this is due to the existence of the commuting diagram
which says that the image of any $(g,s) \in O(q)_+ \wedge S^q$ in the quotient $Q(q)_+ \wedge_{Q(q-k)} S^q$ is labeled by $([g],s)$.
It follows that $(\Sigma^n\lambda_n)_q$ is the smash product of a projection map of coset spaces with the identity on the sphere:
Now finally observe that this projection function
is $(q - n -1 )$-connected (see here). Hence its smash product with $S^q$ is $(2q - n -1 )$-connected.
The key here is the fast growth of the connectivity with $q$. This implies that for each $s$ there exists $q$ such that $\pi_{s+q}((\Sigma^n \lambda_n)_q)$ becomes an isomorphism. Hence $\Sigma^n \lambda_n$ is a stable weak homotopy equivalence and therefore so is $\lambda_n$.
symmetric case
Here the morphism $\lambda_n$ 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
does not become highly connected as $q$ increases, due to the discrete topological space underlying the symmetric group. Accordingly the conclusion now is the opposite: $\lambda_n$ 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.
For $A, B \in Top^{\ast/}$ and for $k,\ell \in \mathbb{N}$, 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
(MMSS 00, lemma 1.8, lemma 21.3)
Consider the following sequence of natural isomorphisms
where we used the adjoint characterization (here) of the Day convolution. Since this is natural in $Z$, the Yoneda lemma implies the claim.
The symmetric monoidal smash product of spectra of the free spectrum constructions (def. 6) on the generating cofibrations $\{S^{n-1}\overset{i_n}{\hookrightarrow} D^n\}_{n \in \mathbb{B}}$ of the classical model structure on topological spaces is given by addition of indices
By lemma 7 the commuting diagram defining the pushout product of free spectra
is equivalent to this diagram:
Since the free spectrum construction is a left adjoint, it preserves pushouts, and so
where in the second step we used this lemma.
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.
Here we discuss model structures for plain spectra, below we discuss model structures for ring spectra and module spectra.
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 $\mathbb{S}_{dia} Mod_{stable}$ 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
Moreover $\mathbb{S}_{dia} Mod_{stable}$ is a
$Top^{\ast/}_{Quillen}$-enriched model category.
Specifically, this specializes to
fro $Dia = Seq$: the standard model structure on topological sequential spectra;
for $Dia = Sym$: the standard standard model structure on symmetric spectra;
for $Dia = Orth$: the standard topological model structure on orthogonal spectra;
for $Dia = Top^{\ast/}_{CW}$: the topological model structure for excisive functors;
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.
By construction, the morphisms $\lambda_n$ of lemma 4 are stable equivalences according to def. 5.
We need to in addition resolve them by suitable cofibrations:
For $n \in \mathbb{N}$ let
be a factorization of the morphism $\lambda_n$ 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:
Recall the sets
of generating cofibrations and generating acyclic cofibrations, respectively, of the classical model structure on topological spaces.
Write
for the class of free spectra, def. 6, on the class $I$ 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 $\mathbb{S}_{dia} FreeMod$ (the site for $\mathbb{S}_{dia}Mod$ from lemma 2).
Similarly, write
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
for the disjoint union of $F J$ with the pushout products of the resolved maps $k_n$ from def. 8 with the elements in $I$.
The sets $F I$ and $F J$ from def. 9 (disregarding the set $K$ there) are, respectively sets of generating cofibrations and generating acyclic cofibrations for the strict model structure $\mathbb{S}_{Dia}Mod_{strict}$ (prop. 4).
By prop. 4 the strict model structure is equivalently the projective pointed model structure on topologically enriched functors
With this the statement follows by the proof of this theorem.
Every element in $K$ (def. 9) is both:
First regarding strict cofibrations: By the Yoneda lemma, the elements in $F J$ have right lifting property against the strict fibrations, hence in particular they are strict cofibrations. Moreover, by Joyal-Tierney calculus, $k_n \Box i_+$ has left lifting against any acyclic strict fibration $f$ precisely if $k_n$ has left lifting against $f^i$. By $\mathbb{S}_{dia} Mod_{strict}$ behaving like a $Top$-enriched model category for one argument a relative CW-complex, the latter is still a strict acyclic fibration. Since $k_n$ by construction is a strict cofibration, the lifting follows and hence also $k_n \Box i_+$ is a strict cofibration.
Regarding stable equivalences: The morphisms in $F J$ by design are strict weak equivalences, hence they are in particular stable equivalences. Similarly, the morphisms $k_n$ by construction, by two-out-of-three and by remark 5 are stable equivalences. Hence the derived hom (…expand…) out of $k_n \Box i_+$ 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 $K$ in def. 9 is to make the following true:
A morphism $f \colon X \to Y$ in $\mathbb{S}_{dia} Mod$ is a $K$-injective morphism (for $K$ from def. 9) precisely if
it is a fibration in the strict model structure (hence degreewise a fibration)
for all $n \in \mathbb{N}$ the commuting squares of structure map compatibility on the underlying sequential spectra
exhibit homotopy pullbacks.
By prop 7, lifting against $F J$ alone characterizes strict fibrations, hence degreewise fibrations. Lifting against the remaining pushout product morphism $k_n \Box i_+$ is, by Joyal-Tierney calculus, equivalent to left lifting $i_+$ against the dual pullback product of $f^{k_n}$, which means that $f^{k_n}$ is a weak homotopy equivalence. But by construction (lemma 4) $f^{k_n}$ is the comparison morphism into the homotopy pullback under consideration.
The $K$-injective objects (for $K$ from def. 9) are precisely the Omega-spectra, def. 5.
A morphism in $\mathbb{S}_{dia}Mod$ which is both
a stable equivalence (def. 5);
a $K$-injective morphisms (with respect to $K$ 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;
Let $f\colon X \to B$ be both a stable equivalence as well as a $K$-injective morphism. Since $K$ contains, by prop. 7, the generating acyclic cofibrations for the strict model structure of prop. 4, $f$ is in particular a strict fibration, hence a degreewise fibration. Therefore the fiber $F$ of $f$ is its homotopy fiber in the strict model structure.
We now want to claim that:
It follows that $F \to \ast$ is a stable equivalence.
This needs some explanation:
Consider for any $E \in \mathbb{S}_{dia}Mod$ the cofiber sequence
$[B,E]_{strict} \overset{p^\ast}{\longrightarrow} [X,E]_{strict} \overset{}{\longrightarrow} [hocof(p),E]_{strict} \longrightarrow [\Sigma B, E]_{strict} \overset{\Sigma p^\ast}{\longrightarrow}[\Sigma X,E]_{strict}$
This kind of sequence is long exact for every pointed model category, not necessarily stable. Then let $E$ be any Omega-spectrum. By assumption it follows then that $p^\ast$ and $\Sigma p^\ast$ are isomorphisms, so that exactness implies that $[hocof(p),E]_{strict} = 0$ for all Omega-spectra $E$.
Now use that on underlying sequential spectra there is a stable weak homotopy equivalence $hocof(p) \longrightarrow \Sigma hofib(p) = \Sigma F$ (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 $[\Sigma F, E]_{strict}$ for all Omega-spectra $E$, hence $[F, \Omega E]_{strict} = 0$ for all Omega-spectra.
We are to conclude that hence $F \to \ast$ is a stable equivalence. But to conclude this we now need to know that every Omega-spectrum in $\mathbb{S}_{dia}Mod$ is in the image under $\Omega$ of an Omega-spectrum, up to strict equivalence. Indeed one shows that there is a shift functor $sh$ on structured spectra and that $E \overset{}{\longrightarrow} \Omega sh E$ is degreewise a weak homotopy equivalence.
Observe also that $F$, being the pullback of a $K$-injective morphisms (by the standard closure properties) is a $K$-injective object, so that by corollary 1 $F$ is an Omega-spectrum. Together this implies with prop. 5 that $F \to \ast$ 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 $\pi_{\bullet \geq 1}(f_n)$ is a weak homotopy equivalence for all $n$ and for each homotopy group in positive degree.
To infer from this the remaining case that also $\pi_0(f_0)$ is an isomorphism, observe that, by assumption of $K$-injectivity, lemma 10 gives that $f_n$ is a homotopy pullback (in topological spaces) of $\Omega (f_{n+1})$. But, by the above, $\Omega (f_{n+1})$ is a weak homotopy equivalence, since $\pi_\bullet(\Omega(-)) = \pi_{\bullet+1}(-)$. Therefore $f_n$ is the homotopy pullback of a weak homotopy equivalence and hence itself a weak homotopy equivalence.
For $K$ from def. 9 the retracts of $K$-relative cell complexes are precisely the morphisms which are
Since all elements of $K$ are stable equivalences and strict cofibrations by lemma 9, it follows that every retract of a relative $K$-cell complex has the same property.
In the other direction, if $f$ is a stable equivalence and strict cofibration, by the small object argument it factors $f \colon \stackrel{i}{\to}\stackrel{p}{\to}$ as a relative $K$-cell complex $i$ followed by a $K$-injective morphism $p$. By the previous statement $i$ is a stable equivalence, and so by assumption and by two-out-of-three so is $p$. Therefore lemma 11 implies that $p$ is a strict acyclic fibration. But then the assumption that $f$ is a strict cofibration means that it has the left lifting property against $p$, and so the retract argument implies that $f$ is a retract of the relative $K$-cell complex $i$.
For $K$ from def. 9 the $K$-injective morphisms are precisely those which are injective with respect to the cofibrations of the strict model structure that are also stable equivalences.
A morphism in $\mathbb{S}_{dia}Mod$ is both
a stable equivalence (def. 5)
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).
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 $K$-injective morphism by corollary 2, and hence if it is also a stable equivalence then by lemma 11 it is a strict acylic fibration.
(of theorem 1)
The non-trivial points to check are the two weak factorization systems.
That $(cof_{stable}\cap weq_{stable} \;,\; fib_{stable})$ 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 $(cof_{stable} \;,\; fib_{stable} \cap we_{stable})$ is identified with that of the strict model structure $(cof_{strict} \;,\; fib_{strict} \cap we_{strict})$.
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.
In $\mathbb{S}Mod$, $\mathbb{S}_{Orth} Mod$ and in $\mathbb{S}_{Seq} Mod$ we have for the concepts from def. 5 that
In $\mathbb{S}_{Sym}Mod$ however we only have
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 $K$ of generating acylic cofibrations in def. 9.
For $\lambda_0 \colon F^{dia}_1 S^1 \to F^{dia}_0 S^0$ from lemma 4, write
for the mapping spectrum construction out of $\lambda_1$ into $X$.
Write
for the homotopy colimit over the resulting sequence of iterations (formed with respec to the strict model structure of prop. 4). Write
for the 0th-component map into the colimit.
The functor $R^\infty$ from def. 10 has the following properties.
for $E$ an Omega-spectrum according to def. 5, then, by lemma 4, $\lambda_0^\ast(E)$ is weak equivalence in the strict model structure (prop. 4), and hence so is $r_E$;
for $f\colon X \longrightarrow Y$ a stable weak homotopy equivalence according to def. 5, then $R^\infty f \colon R^\infty X \longrightarrow R^\infty Y$ is a weak equivalence in the strict model structure.
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
and
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)
Let $E$ be an Omega-spectrum. Then by the first item of lemma 14, for every $X$ the morphism
is an isomorphism. Since $r_{(-)}$ is a natural transformation (by def. 10), the naturality squares give a factorization of this morphism as
Combining this with vertical morphisms as below, which are isomorphisms again by item 1 of lemma 14,
exhibits a retraction
which is natural in $X$ (that the bottom and right composite is indeed the identity is again the naturality of $r_{(-)}$). This naturality now implies a retraction of morphisms
Finally, by the second item of lemma 14, the middle vertical morphism here is an isomorphism, hence $[f^\ast, E]_{strict}$ is the retract of an iso and hence (here) an isomorphism itself, for all Omega-spectra $E$. This means by definition that $f$ is a stable equivalence.
Now for the converse.
In the case $Dia \in \{Top^{\ast/}, Orth, Seq\}$, hence for sequential spectra, orthogonal spectra and pre-excisive functors, stable equivalences are stable weak homotopy equivalences (def. 5).
By theorem ref. 1, and by lemmas 11 and 12, every stable weak equivalence factors as a $K$-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 $K$-relative cell complex is a stable weak homotopy equivalence.
Observe now that we know this to be true for the elements of $K$ itself (def. 9): first of all, the elements in $F J$ are retracts of deformation retracts and therefore stable weak homotopy equivalences. Second, the morphisms denoted $k_n$ in def. 9, which resolve the maps $\lambda_n$ from def. 7, are stable weak homotopy equivalences by lemma 6. This implies that so are their pushout products $k_n \Box i$.
This completes the proof of theorem 2.
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
We give the proof below, after a few preliminaries.
The sequence of adjunctions between the categories $\mathbb{S}_{dia}Mod$ from prop. 3 are Quillen adjunctions with respect to the stable model structures of theorem 1
By lemma 12 the stable fibrations are equivalently the $K$-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.
(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 $\mathbb{S}_{Sym}Mod$.
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 $Top^{\ast/}$ and the homotopy lifting property, and since (…)
(…)
(…)
The stable model structures from theorem 1 on the categories of modules from prop. 2, remark 2
are compatible with their monoidal category structure given by the symmetric monoidal smash product of spectra $\wedge$ of def. 3, in that $(\mathbb{S}_{dia} Mod_{stable}, \wedge_{\mathbb{S}_{dia}})$ in these cases
is a stable model category;
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.
The pushout product of two cofibrations in $\mathbb{S}_{dia}Mod_{stable}$ is again a cofibration.
A general abstract fact about pushout-products (Hovey-Shipley-Smith 00, prop. 5.3.4, see here) is that for $I_1, I_2$ two classes of morphisms in a closed symmetric monoidal category with finite limits and finite colimits, and writing $I_i Cof$ for their saturated classes, then under pushout-product $\Box$:
Since the cofibrations of the stable model structure, theorem 1, are elements in
with $F I$ the class of free spectra on the class of generating cofibrations $I$ of the classical model structure on topological spaces, def. 9, this implies in the present case that
Now lemma 8 implies that
and hence the claim follows.
Let $Y \in \mathbb{S}_{dia} Mod_{stable}$ be cofibrant. Then the smash product of spectra-functor (def. 3)
preserves stable weak homotopy equivalences as well as stable equivalences (def. 5).
For every $X \in \mathbb{S}_{dia} Mod$, the functor
sends acylic cofibrations in the stable model structure to morphisms that are stable equivalences and h-cofibrations.
The symmetric monoidal smash product of spectra $\wedge_{\mathbb{S}_{dia}}$ on $\mathbb{S}_{dia} Mod$, def. 3 satisfies the pushout-product axiom with respect to the stable model structure $\mathbb{S}_{dia} Mod$ of theorem 1.
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 (…).
(…)
(…)
(…)
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.
Let $\mathcal{C}, \mathcal{D}$ be pointed topologically enriched categories (def.), i.e. enriched categories over $(Top_{cg}^{\ast/}, \wedge, S^0)$ from example 5.
The pointed topologically enriched opposite category $\mathcal{C}^{op}$ is the topologically enriched category with the same objects as $\mathcal{C}$, with hom-spaces
and with composition given by braiding followed by the composition in $\mathcal{C}$:
the pointed topological product category $\mathcal{C} \times \mathcal{D}$ is the topologically enriched category whose objects are pairs of objects $(c,d)$ with $c \in \mathcal{C}$ and $d\in \mathcal{D}$, whose hom-spaces are the smash product of the separate hom-spaces
and whose composition operation is the braiding followed by the smash product of the separate composition operations:
A pointed topologically enriched functor (def.) into $Top^{\ast/}_{cg}$ (exmpl.) out of a pointed topological product category as in def. 11
(a “pointed topological bifunctor”) has component maps of the form
By functoriallity and under passing to adjuncts (cor.) this is equivalent to two commuting actions
and
In the special case of a functor out of the product category of some $\mathcal{C}$ with its opposite category (def. 11)
then this takes the form
and
Let $\mathcal{C}$ be a small pointed topologically enriched category (def.), i.e. an enriched category over $(Top_{cg}^{\ast/}, \wedge, S^0)$ from example 5. Let
be a pointed topologically enriched functor (def.) out of the pointed topological product category of $\mathcal{C}$ with its opposite category, according to def. 11.
The coend of $F$, denoted $\overset{c \in \mathcal{C}}{\int} F(c,c)$, is the coequalizer in $Top_{cg}^{\ast}$ (prop., exmpl., prop., cor.) of the two actions encoded in $F$ via example 2:
The end of $F$, denoted $\underset{c\in \mathcal{C}}{\int} F(c,c)$, is the equalizer in $Top_{cg}^{\ast/}$ (prop., exmpl., prop., cor.) of the adjuncts of the two actions encoded in $F$ via example 2:
Let $\mathcal{C}$ be a small pointed topologically enriched category (def.). For $F,G \;\colon\; \mathcal{C} \longrightarrow Top^{\ast/}_{cg}$ two pointed topologically enriched functors, then the end (def. 12) of $Maps(F(-),G(-))_\ast$ is a topological space whose underlying pointed set is the pointed set of natural transformations $F\to G$ (def.)
The underlying pointed set functor $U\colon Top^{\ast/}_{cg}\to Set^{\ast/}$ preserves all limits (prop., prop., prop.). Therefore there is an equalizer diagram in $Set^{\ast/}$ of the form
Here the object in the middle is just the set of collections of component morphisms $\left\{ F(c)\overset{\eta_c}{\to} G(c)\right\}_{c\in \mathcal{C}}$. The two parallel maps in the equalizer diagram take such a collection to the functions which send any $c \overset{f}{\to} d$ to the result of precomposing
and of postcomposing
each component in such a collection, respectively. These two functions being equal, hence the collection $\{\eta_c\}_{c\in \mathcal{C}}$ being in the equalizer, means precisley that for all $c,d$ and all $f\colon c \to d$ the square
is a commuting square. This is precisley the condition that the collection $\{\eta_c\}_{c\in \mathcal{C}}$ be a natural transformation.
Conversely, example 3 says that ends over bifunctors of the form $Maps(F(-),G(-)))_\ast$ constitute hom-spaces between pointed topologically enriched functors:
Let $\mathcal{C}$ be a small pointed topologically enriched categories (def.). Define the structure of a pointed topologically enriched category on the category $[\mathcal{C}, Top_{cg}^{\ast/}]$ of pointed topologically enriched functors to $Top^{\ast/}_{cg}$ (exmpl.) by taking the hom-spaces to be given by the ends (def. 12) of example 3:
and by taking the composition maps to be the morphisms induced by the maps
by observing that these equalize the two actions in the definition of the end.
The resulting pointed topologically enriched category $[\mathcal{C},Top^{\ast/}_{cg}]$ is also called the $Top^{\ast/}_{cg}$-enriched functor category over $\mathcal{C}$ with coefficients in $Top^{\ast/}_{cg}$.
First of all this yields a concise statement of the pointed topologically enriched Yoneda lemma (prop.)
(topologically enriched Yoneda lemma)
Let $\mathcal{C}$ be a small pointed topologically enriched categories (def.). For $F \colon \mathcal{C}\to Top^{\ast/}_{cg}$ a pointed topologically enriched functor (def.) and for $c\in \mathcal{C}$ an object, there is a natural isomorphism
between the hom-space of the pointed topological functor category, according to def. 13, from the functor represented by $c$ to $F$, and the value of $F$ on $c$.
In terms of the ends (def. 12) defining these hom-spaces, this means that
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:
Let $\mathcal{C}$ be a small pointed topologically enriched categories (def.). For $F \colon \mathcal{C}\to Top^{\ast/}_{cg}$ a pointed topologically enriched functor (def.) and for $c\in \mathcal{C}$ an object, there is a natural isomorphism
Moreover, the morphism that hence exhibits $F(c)$ as the coequalizer of the two morphisms in def. 12 is componentwise the canonical action
which is adjunct to the component map $\mathcal{C}(d,c) \to Maps(F(c),F(d))_{\ast}$ of the topologically enriched functor $F$.
(e.g. MMSS 00, lemma 1.6)
The coequalizer of pointed topological spaces that we need to consider has underlying it a coequalizer of underlying pointed sets (prop., prop., prop.). That in turn is the colimit over the diagram of underlying sets with the basepointe adjoined to the diagram (prop.). For a coequalizer diagram adding that extra point to the diagram clearly does not change the colimit, and so we need to consider the plain coequalizer of sets.
That is just the set of equivalence classes of pairs
where two such pairs
are regarded as equivalent if there exists
such that
(Because then the two pairs are the two images of the pair $(g,x)$ under the two morphisms being coequalized.)
But now considering the case that $d = c_0$ and $d = id_{c_0}$, so that $f = \phi$ shows that any pair
is identified, in the coequalizer, with the pair
hence with $\phi(x)\in F(c_0)$.
This shows the claim at the level of the underlying sets. To conclude it is now sufficient (prop.) to show that the topology on $F(c_0) \in Top^{\ast/}_{cg}$ is the final topology (def.) of the system of component morphisms
which we just found. But that system includes
which is a retraction
and so if all the preimages of a given subset of the coequalizer under these component maps is open, it must have already been open in $F(c)$.
The statement of the co-Yoneda lemma in prop. 12 is a kind of categorification of the following statement in analysis (whence the notation with the integral signs):
For $X$ a topological space, $f \colon X \to\mathbb{R}$ a continuous function and $\delta(-,x_0)$ denoting the Dirac distribution, then
It is this analogy that gives the name to the following statement:
(Fubini theorem for (co)-ends)
For $F$ a pointed topologically enriched bifunctor on a small pointed topological product category $\mathcal{C}_1 \times \mathcal{C}_2$ (def. 11), i.e.
then its end and coend (def. 12) is equivalently formed consecutively over each variable, in either order:
and
Because the pointed compactly generated mapping space functor (exmpl.)
takes colimits in the first argument and limits in the second argument to limits (cor.), it also takes coends in the first argument and ends in the second argument, to ends (def. 12):
and
(left Kan extension via coends)
Let $\mathcal{C}, \mathcal{D}$ be small pointed topologically enriched categories (def.) and let
be a pointed topologically enriched functor (def.). Then precomposition with $p$ constitutes a functor
$G\mapsto G\circ p$. This functor has a left adjoint $Lan_p$, called left Kan extension along $p$
which is given objectwise by a coend (def. 12):
Use the expression of natural transformations in terms of ends (example 3 and def. 13), then use the respect of $Maps(-,-)_\ast$ 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:
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.
A (pointed) topologically enriched monoidal category is a (pointed) topologically enriched category $\mathcal{C}$ (def.) equipped with
a (pointed) topologically enriched functor (def.)
out of the (pointed) topologival product category of $\mathcal{C}$ with itself (def. 11), called the tensor product,
an object
called the unit object or tensor unit,
called the associator,
called the left unitor, and a natural isomorphism
called the right unitor,
such that the following two kinds of diagrams commute, for all objects involved:
triangle identity:
the pentagon identity:
A (pointed) topological braided monoidal category, is a (pointed) topological monoidal category $\mathcal{C}$ (def. 14) equipped with a natural isomorphism
called the braiding, such that the following two kinds of diagrams commute for all objects involved:
and
where $a_{x,y,z} \colon (x \otimes y) \otimes z \to x \otimes (y \otimes z)$ denotes the components of the associator of $\mathcal{C}^\otimes$.
A (pointed) topological symmetric monoidal category is a (pointed) topological braided monoidal category (def. 15) for which the braiding
satisfies the condition:
for all objects $x, y$
Given a (pointed) topological symmetric monoidal category $\mathcal{C}$ with tensor product $\otimes$ (def. 16) it is called a closed monoidal category if for each $X\in \mathcal{X}$ the functor $X \otimes(-)\simeq (-)\otimes X$ has a right adjoint, denoted
For any other object $Y$, the object $[X,Y] \in \mathcal{C}$ is then called the internal hom object between $X$ and $Y$.
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 $Top_{cg}$ of compactly generated topological spaces (def.) becomes a symmetric monoidal category with tensor product the corresponding Cartesian products, hence the operation of forming k-ified (cor.) product topological spaces (exmpl.). The underlying functions of the associator, unitor and braiding isomorphisms are just those of the underlying sets, as above.
Symmetric monoidal categories, such as these, for which the tensor product is the Cartesian product are called Cartesian monoidal categories.
The category $Top_{cg}^{\ast/}$ of pointed compactly generated topological spaces with tensor product the smash product $\wedge$ (def.)
is a symmetric monoidal category (def. 16) with unit object the pointed 0-sphere $S^0$.
The components of the associator, the unitors and the braiding are those of Top as in example 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..
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:
A monoid in $(Ab, \otimes_{\mathbb{Z}}, \mathbb{Z})$ (def. 19) is equivalently a ring.
A commutative monoid in in $(Ab, \otimes_{\mathbb{Z}}, \mathbb{Z})$ (def. 19) is equivalently a commutative ring $R$.
An $R$-module object in $(Ab, \otimes_{\mathbb{Z}}, \mathbb{Z})$ (def. 20) is equivalently an $R$-module;
The tensor product of $R$-module objects (def. 21) is the standard tensor product of modules.
The category of module objects $R Mod(Ab)$ (def. 21) is the standard category of modules $R Mod$.
Let $(\mathcal{C},\otimes_{\mathcal{C}}, 1_{\mathcal{C}})$ and $(\mathcal{D},\otimes_{\mathcal{D}}, 1_{\mathcal{D}} )$ be two (pointed) topologically enriched monoidal categories (def. 14). A topologically enriched lax monoidal functor between them is
a topologically enriched functor
a morphism
for all $x,y \in \mathcal{C}$
satisfying the following conditions:
(associativity) For all objects $x,y,z \in \mathcal{C}$ the following diagram commutes
where $a^{\mathcal{C}}$ and $a^{\mathcal{D}}$ denote the associators of the monoidal categories;
(unitality) For all $x \in \mathcal{C}$ the following diagrams commutes
and
where $\ell^{\mathcal{C}}$, $\ell^{\mathcal{D}}$, $r^{\mathcal{C}}$, $r^{\mathcal{D}}$ denote the left and right unitors of the two monoidal categories, respectively.
If $\epsilon$ and alll $\mu_{x,y}$ are isomorphisms, then $F$ is called a strong monoidal functor.
If moreover $(\mathcal{C},\otimes_{\mathcal{C}}, 1_{\mathcal{C}})$ and $(\mathcal{D},\otimes_{\mathcal{D}}, 1_{\mathcal{D}} )$ are equipped with the structure of braided monoidal categories (def. 15), then the lax monoidal functor $F$ is called a braided monoidal functor if in addition the following diagram commutes for all objects $x,y \in \mathcal{C}$
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 $(\mathcal{C},\otimes_{\mathcal{C}}, 1_{\mathcal{C}})$ and $(\mathcal{D},\otimes_{\mathcal{D}}, 1_{\mathcal{D}} )$ are symmetric monoidal categories (def. 16) then a braided monoidal functor (def. 18) between them is often called a symmetric monoidal functor.
Given a (pointed) topological monoidal category $(\mathcal{C}, \otimes, 1)$, then a monoid internal to $(\mathcal{C}, \otimes, 1)$ is
an object $A \in \mathcal{C}$;
a morphism $e \;\colon\; 1 \longrightarrow A$ (called the unit)
a morphism $\mu \;\colon\; A \otimes A \longrightarrow A$ (called the product);
such that
(associativity) the following diagram commutes
where $a$ is the associator isomorphism of $\mathcal{C}$;
(unitality) the following diagram commutes:
where $\ell$ and $r$ are the left and right unitor isomorphisms of $\mathcal{C}$.
Moreover, if $(\mathcal{C}, \otimes , 1)$ has the structure of a symmetric monoidal category (def. 16) $(\mathcal{C}, \otimes, 1, B)$ with symmetric braiding $\tau$, then a monoid $(A,\mu, e)$ as above is called a commutative monoid in $(\mathcal{C}, \otimes, 1, B)$ if in addition
(commutativity) the following diagram commutes
A homomorphism of monoids $(A_1, \mu_1, e_1)\longrightarrow (A_2, \mu_2, f_2)$ is a morphism
in $\mathcal{C}$, such that the following two diagrams commute
and
Write $Mon(\mathcal{C}, \otimes,1)$ for the category of monoids in $\mathcal{C}$, and $CMon(\mathcal{C}, \otimes, 1)$ for its subcategory of commutative monoids.
Given a (pointed) topological monoidal category $(\mathcal{C}, \otimes, 1)$, and given $(A,\mu,e)$ a monoid in $(\mathcal{C}, \otimes, 1)$ (def. 19), then a left module object in $(\mathcal{C}, \otimes, 1)$ over $(A,\mu,e)$ is
an object $N \in \mathcal{C}$;
a morphism $\rho \;\colon\; A \otimes N \longrightarrow N$ (called the action);
such that
(unitality) the following diagram commutes:
where $\ell$ is the left unitor isomorphism of $\mathcal{C}$.
(action property) the following diagram commutes
A homomorphism of left $A$-module objects
is a morphism
in $\mathcal{C}$, such that the following diagram commutes:
For the resulting category of modules of left $A$-modules in $\mathcal{C}$ with $A$-module homomorphisms between them, we write
This is naturally a (pointed) topologically enriched category itself.
Given a (pointed) topological symmetric monoidal category $(\mathcal{C}, \otimes, 1)$ (def. 16), given $(A,\mu,e)$ a commutative monoid in $(\mathcal{C}, \otimes, 1)$ (def. 19), and given $(N_1, \rho_1)$ and $(N_2, \rho_2)$ two left $A$-module objects (def.19), then the tensor product of modules $N_1 \otimes_A N_2$ is, if it exists, the coequalizer
Given a (pointed) topological symmetric monoidal category $(\mathcal{C}, \otimes, 1)$ (def. 16), and given $(A,\mu,e)$ a commutative monoid in $(\mathcal{C}, \otimes, 1)$ (def. 19). If all coequalizers exist in $\mathcal{C}$, then the tensor product of modules $\otimes_A$ from def. 21 makes the category of modules $A Mod(\mathcal{C})$ into a symmetric monoidal category, $(A Mod, \otimes_A, A)$ with tensor unit the object $A$ itself.
Given a monoidal category of modules $(A Mod , \otimes_A , A)$ as in prop. 15, then a monoid $(E, \mu, e)$ in $(A Mod , \otimes_A , A)$ (def. 19) is called an $A$-algebra.
Given a monoidal category of modules $(A Mod , \otimes_A , A)$ in a monoidal category $(\mathcal{C},\otimes, 1)$ as in prop. 15, and an $A$-algebra $(E,\mu,e)$ (def. 22), then there is an equivalence of categories
between the category of commutative monoids in $A Mod$ and the coslice category of commutative monoids in $\mathcal{C}$ under $A$, hence between commutative $A$-algebras in $\mathcal{C}$ and commutative monoids $E$ in $\mathcal{C}$ that are equipped with a homomorphism of monoids $A \longrightarrow E$.
(e.g. EKMM 97, VII lemma 1.3)
In one direction, consider a $A$-algebra $E$ with unit $e_E \;\colon\; A \longrightarrow E$ and product $\mu_{E/A} \colon E \otimes_A E \longrightarrow E$. There is the underlying product $\mu_E$
By considering a diagram of such coequalizer diagrams with middle vertical morphism $e_E\circ e_A$, one find that this is a unit for $\mu_E$ and that $(E, \mu_E, e_E \circ e_A)$ is a commutative monoid in $(\mathcal{C}, \otimes, 1)$.
Then consider the two conditions on the unit $e_E \colon A \longrightarrow E$. First of all this is an $A$-module homomorphism, which means that
commutes. Moreover it satisfies the unit property
By forgetting the tensor product over $A$, the latter gives
where the top vertical morphisms on the left the canonical coequalizers, which identifies the vertical composites on the right as shown. Hence this may be pasted to the square $(\star)$ above, to yield a commuting square
This shows that the unit $e_A$ is a homomorphism of monoids $(A,\mu_A, e_A) \longrightarrow (E, \mu_E, e_E\circ e_A)$.
Now for the converse direction, assume that $(A,\mu_A, e_A)$ and $(E, \mu_E, e'_E)$ are two commutative monoids in $(\mathcal{C}, \otimes, 1)$ with $e_E \;\colon\; A \to E$ a monoid homomorphism. Then $E$ inherits a left $A$-module structure by
By commutativity and associativity it follows that $\mu_E$ coequalizes the two induced morphisms $E \otimes A \otimes E \underoverset{\longrightarrow}{\longrightarrow}{\phantom{AA}} E \otimes E$. Hence the universal property of the coequalizer gives a factorization through some $\mu_{E/A}\colon E \otimes_A E \longrightarrow E$. This shows that $(E, \mu_{E/A}, e_E)$ is a commutative $A$-algebra.
Finally one checks that these two constructions are inverses to each other, up to isomorphism.
Let $\mathcal{C}$ be a small pointed topological monoidal category (def. 14) with tensor product denoted $\otimes_{\mathcal{C}} \;\colon\; \mathcal{C} \times\mathcal{C} \to \mathcal{C}$.
Then the Day convolution tensor product on the pointed topological enriched functor category $[\mathcal{C},Top^{\ast/}_{cg}]$ (def. 13) is the functor
out of the pointed topological product category (def. 11) given by the following coend (def. 12)
Let $Seq$ denote the category with objects the natural numbers, and only the zero morphisms and identity morphisms on these objects:
Regard this as a pointed topologically enriched category in the unique way. The operation of addition of natural numbers $\otimes = +$ makes this a monoidal category.
An object $X_\bullet \in [Seq, Top_{cg}^{\ast/}]$ is an $\mathbb{N}$-sequence of pointed topological spaces. Given two such, then their Day convolution according to def. 23 is
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.
Let $\mathcal{C}$ be a small pointed topologically enriched category (def.). Its external tensor product is the pointed topologically enriched functor
given by
i.e.
The Day convolution product (def. 23) of two functors is equivalently the left Kan extension (def. 14) of their external tensor product (def. 24) along the tensor product $\otimes_{\mathcal{C}}$: there is a natural isomorphism
Hence the adjunction unit is a natural transformation of the form
This perspective is highlighted in (MMSS 00, p. 60).
By prop. 14 we may compute the left Kan extension as the following coend:
The Day convolution $\otimes_{Day}$ (def. 23) is universally characterized by the property that there are natural isomorphisms
where $\overline{\wedge}$ is the external product of def. 24.
Write
for the $Top^{\ast/}_{cg}$-Yoneda embedding, so that for $c\in \mathcal{C}$ any object, $y(c)$ is the corepresented functor $y(c)\colon d \mapsto \mathcal{C}(c,d)$.
For $\mathcal{C}$ a small pointed topological monoidal category (def. 14), the Day convolution tensor product $\otimes_{Day}$ of def. 23 makes the pointed topologically enriched functor category
a pointed topological monoidal category (def. 14) with tensor unit $y(1)$ co-represented by the tensor unit $1$ of $\mathcal{C}$.
Regarding associativity, observe that
where we used the Fubini theorem for coends (prop. 13) and then twice the co-Yoneda lemma (prop. 12). An analogous formula follows for $X \otimes_{Day} (Y \otimes_{Day} Z)))(c)$, and so associativity follows via prop. 17 from the associativity of the smash product and of the tensor product $\otimes_{\mathcal{C}}$.
To see that $y(1)$ is the tensor unit for $\otimes_{Day}$, use the Fubini theorem for coends (prop. 13) and then twice the co-Yoneda lemma (prop. 12) to get for any $X \in [\mathcal{C},Top^{\ast/}_{cg}]$ that
For $\mathcal{C}$ a small pointed topological monoidal category (def. 14) with tensor product denoted $\otimes_{\mathcal{C}} \;\colon\; \mathcal{C} \times\mathcal{C} \to \mathcal{C}$, the monoidal category with Day convolution $([\mathcal{C},Top^{\ast/}_{cg}], \otimes_{Day}, y(1))$ from def. 18 is a closed monoidal category (def. 17). Its internal hom $[-,-]_{Day}$ is given by the end (def. 12)
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:
In the situation of def. 18, the Yoneda embedding $c\mapsto \mathcal{C}(c,-)$ constitutes a strong monoidal functor (def. 18)
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
Let $(\mathcal{C},\otimes I)$ be a pointed topologically enriched category (symmetric monoidal category) monoidal category (def. 14). Regard $(Top_{cg}^{\ast/}, \wedge , S^0)$ as a topological symmetric monoidal category as in example 5.
Then (commutative) monoids in (def. 19) the Day convolution monoidal category $([\mathcal{C}, Top^{\ast/}_{cg}], \otimes_{Day}, y(1_{\mathcal{C}}))$ of prop. 18 are equivalent to (braided) lax monoidal functors (def. 18) of the form
called functors with smash products on $\mathcal{C}$, i.e. there are equivalences of categories of the form
Moreover, module objects over these monoid objects are equivalent to the corresponding modules over monoidal functors.
This is stated in some form in (Day 70, example 3.2.2). It is highlighted again in (MMSS 00, prop. 22.1).
By definition 18, a lax monoidal functor $F \colon \mathcal{C} \to Top^{\ast/}_{cg}$ is a topologically enriched functor equipped with a morphism of pointed topological spaces of the form
and equipped with a natural system of maps of pointed topological spaces of the form
for all $c_1,c_2 \in \mathcal{C}$.
Under the Yoneda lemma (prop. 11) the first of these is equivalently a morphism in $[\mathcal{C}, Top^{\ast/}_{cg}]$ of the form
Moreover, under the natural isomorphism of corollary 3 the second of these is equivalently a morphism in $[\mathcal{C}, Top^{\ast/}_{cg}]$ of the form
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 $F$ under $\otimes_{Day}$.
Similarly for module objects and modules over monoidal functors.
Let $f \;\colon\; \mathcal{C} \longrightarrow \mathcal{D}$ be a lax monoidal functor (def. 18) between pointed topologically enriched monoidal categories (def. 14). Then the induced functor
given by $(f^\ast X)(c)\coloneqq X(f(c))$ preserves monoids under Day convolution
Moreover, if $\mathcal{C}$ and $\mathcal{D}$ are symmetric monoidal categories (def. 16) and $f$ is a braided monoidal functor (def. 18), then $f^\ast$ also preserves commutative monoids
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.
We now combine the concepts of model category and monoidal category.
A (symmetric) monoidal model category is model category $\mathcal{C}$ equipped with the structure of a closed symmetric monoidal category $(\mathcal{C}, \otimes, I)$ such that the following two compatibility conditions are satisfied
(pushout-product axiom) For every pair of cofibrations $f \colon X \to Y$ and $f' \colon X' \to Y'$, their pushout-product, hence the induced morphism out of the cofibered coproduct over ways of forming the tensor product of these objects
is itself a cofibration, which, furthermore, is acyclic if $f$ or $f'$ is.
(Equivalently this says that the tensor product $\otimes : C \times C \to C$ is a left Quillen bifunctor.)
(unit axiom) For every cofibrant object $X$ and every cofibrant resolution $\emptyset \hookrightarrow Q I \stackrel{p_I}{\longrightarrow} \ast$ of the tensor unit $I$, the resulting morphism
is a weak equivalence.
The pushout-product axiom in def. 25 implies that for $X$ a cofibrant object, then the functor $X \otimes (-)$ preserves cofibrations and acyclic cofibrations.
In particular if the tensor unit $I$ happens to be cofibrant, then the unit axiom in def. 25 is implied by the pushout-product axiom.
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 $X\otimes f$ of acyclic cofibrations $f$ with any object $X$ is a weak equivalence.
(Schwede-Shipley 00, def. 3.3.).
In particular, the axiom in def. 26 says that for every object $X$ the functor $X \otimes (-)$ sends acyclic cofibrations to weak equivalences.
Let $(\mathcal{C}, \otimes, I)$ be a monoidal model category. Then the left derived functor of the tensor product exsists and makes the homotopy category into a monoidal category $(Ho(\mathcal{C}), \otimes^L, \gamma(I))$.
If in in addition $(\mathcal{C}, \otimes)$ satisfies the monoid axiom, then the localization functor $\gamma\colon \mathcal{C}\to Ho(\mathcal{C})$ carries the structure of a lax monoidal functor
Consider the explicit model of $Ho(\mathcal{C})$ as the category of fibrant-cofibrant objects in $\mathcal{C}$ 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 $\otimes^L$ exists and its associativity follows simply by restriction. It remains to see its unitality.
To that end, consider the construction of the localization functor $\gamma$ via a fixed but arbitrary choice of (co-)fibrant replacements $Q$ and $R$, assumed to be the identity on (co-)fibrant objects. We fix notation as follows:
Now to see that $\gamma(I)$ is the tensor unit for $\otimes^L$, notice that in the zig-zag
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 $\gamma(-)$ this zig-zig gives an isomorphism
and similarly for tensoring with $\gamma(I)$ from the right.
To exhibit lax monoidal structure on $\gamma$, we need to construct a natural transformation
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
To this end, consider the zig-zag
and observe that the two morphisms on the left are weak equivalences, as indicated, by the pushout-product axiom satisfied by $\otimes$.
Hence applying $\gamma$ to this zig-zag, which is given by the two horizontal part of the following digram
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 $(R Q Z)$ and consider the following pasting composite:
Observe that under $\gamma$ the total top zig-zag in this diagram gives
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 $\gamma$ 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
The same argument applies to left tensoring with $R Q Z$ 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 $\gamma$ as a lax monoidal functor.