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
The term simplicial model category is short for sSet${}_{Quillen}$-enriched model category.
A simplicial model category is a model or presentation for an (∞,1)-category that is half way in between a bare model category and a Kan complex-enriched category.
Specifically, a simplicial model category is an sSet-enriched category $C$ together with the structure of a model category on its underlying category $C_0$ such that both structures are compatible in a reasonable way.
One important use of simplicial model categories comes from the fact that the full sSet-subcategory $C^\circ \hookrightarrow C$ on the fibrant-cofibrant objects – which is not just sSet-enriched but actually Kan complex-enriched – is the (∞,1)-category-enhancement of the homotopy category of the model category $C_0$.
For generalizations of this construction with sSet replaced by another monoidal model category see enriched homotopical category.
The term simplicial model category for the notion described here is entirely standard, but in itself a bit suboptimal. More properly one would speak of simplicially enriched category, which is a proper special case of a simplicial object in Cat (that for which the simplicial set of objects is discrete).
The other caveat is that there are different model category structures on sSet and hence even the term $sSet$-enriched model category is ambiguous.
For instance the model structure for quasi-categories is an $sSet$-enriched model category, but not for the standard Quillen model structure on the enriching category: since $sSet_{Joyal}$ is a closed monoidal model category it is enriched over itself, hence is a $sSet_{Joyal}$-enriched model category, not an $sSet_{Quillen}$-enriched one. So in the standard terminology, $sSet_{Joyal}$ is not a “simplicial model category”.
A simplicial model category is an enriched model category which is enriched over $sSet_{Quillen}$: the category sSet equipped with its standard model structure on simplicial sets.
Spelled out, this means that a simplicial model category is
with the structure of a model category on the underlying category $C_0$
such that for every cofibration $i : A \to B$ and every fibration $p : X \to Y$ in $C_0$ the morphism of simplicial sets $C(B,X) \stackrel{i^* \times p_*}{\to} C(A,X) \times_{C(A,Y)} C(B,Y)$ is a fibration;
Let $\mathcal{C}$ be a category equipped with the structure of a model category and with that of an sSet-enriched category with is tensored and cotensored over sSet.
The following conditions – that each make $\mathcal{C}$ into a simplicial model category – are equivalent:
the tensoring $\otimes : \mathcal{C} \times sSet \to \mathcal{C}$ is a left Quillen bifunctor;
for any cofibration $X \to Y$ and fibration $A \to B$ in $\mathcal{C}$, the induced morphism
is a fibration, and is in addition a weak equivalence if either of the two morphisms is;
for any cofibration $X \to Y$ in $sSet$ and fibration $A \to B$ in $\mathcal{C}$, the induced morphism
is a fibration, and is in addition a weak equivalence if either of the two morphisms is.
This follows directly from the defining properties of tensoring and cotensoring.
We list in the following some implications of these equivalent conditions.
Let $\mathcal{C}$ now be a simplicial model category.
If $A \in \mathcal{C}$ is fibrant, and $X \hookrightarrow Y$ is a cofibration in sSet, then
is a fibration in $\mathcal{C}$.
Apply prop. 1 to the case of the cofibration $X \to Y$ and the fibration $A \to *$, where “$*$” denotes the terminal object. This yields that
is a fibration. But ${*}^Y = {*}^X = {*}$ and hence the claim follows.
Similarly we have
If $X \in \mathcal{C}$ is cofibrant and $A \in \mathcal{C}$ is fibrant, then $\mathcal{C}(Y,X)$ is fibrant in sSet, hence is a Kan complex.
Apply prop. 1 to the cofibration $\emptyset \to X$, where “$\emptyset$” denotes the initial object, and to the fibration $A \to *$ to find that
is a fibration. But since $\emptyset$ is initial and $*$ is terminal, all three simplicial sets in the fiber product on the right are the point, hence this is a fibration
For $X$ and $A$ any two objects and $Q X$ and $P A$ a cofibrant and fibrant replacement, respectively, $\mathcal{C}(Q X, P A)$ is the correct derived hom-space between $X$ and $A$ (see the discussion there). In particular the full $sSet$-enriched subcategory on cofibrant fibrant objects is therefore an sSet-enriched category which is fibrant in the model structure on simplicially enriched categories. Its homotopy coherent nerve is a quasi-category. All this are intrinsic incarnatons of the (∞,1)-category that is presented by $C$.
The classical model structure on simplicial sets $sSet_{Quillen}$ is a closed monoidal model category and is hence naturally enriched, as a model category, over itself. This is the archetypical simplicial model category.
The classical model structure on topological spaces for compactly generated topological spaces (here) is similarly enriched over itself. Under geometric realization this makes also makes it a simplicial model category.
(See also for instance Goerss-Schemmerhorn 06, p. 26)
For $C$ any small sSet-enriched category and $A$ simplicial combinatorial model category, the global model structure on functors $[C^{op}, A]_{proj}$ and $[C^{op},A]_{inj}$ are themselved simplicial combinatorial model categories. See model structure on simplicial presheaves.
The left Bousfield localization of a combinatorial simplicial model category at any set of morphisms is again a combinatorial simplicial model category. Large classes of examples arise this way.
While many model categories do not admit an $sSet_{Quillen}$-enrichment, for large classes of model categories one can find a Quillen equivalence to a model category that does have an $sSet_{Quillen}$-enrichment.
These are constructed as Bousfield localization of Reedy model structures on the category of simplicial objects in the given model category.
Let $C$ be a
By the discussion at cofibrantly generated model category in the section Presentation and generation there exists a small set $E \subset Obj(C)$ of objects that detect weak equivalences. For some such choice of $E$, let
where $e \cdot \Delta[k] : [n] \mapsto \coprod_{\Delta([n],[k])} e$.
Write
for the left Bousfield localization of the projective model structure on functors at this set $S$ of morphisms.
Similarly, write
for the left Bousfield localization of the Reedy model structure at $S$.
Let $C$ be a cofibrantly generated model category.
If $X \in [\Delta^{op}, C]$ is degreewise cofibrant and has all structure maps being weak equivalences, then all $X_i \to hocolim X$ are weak equivalences.
Hence $X \to const\,hocolim X$ is a weak equivalence.
This appears as (Dugger, prop. 5.4 corollary 5.5).
The model structures from def. 2 have the following properties.
The weak equivalences in both are precisely those morphisms which become weak equivalences under homotopy colimit over $\Delta^{op}$.
The fibrant objects in both are precisely those objects that are fibrant in the corresponding unlocalized structures, and such that all the face and degeneracy maps are weak equivalences in $C$.
The colimit/constant adjoint functors
constitute a Quillen equivalence, the identity functors constitute a Quillen equivalence
and the constant/limit adjoint functors (since $\Delta^{op}$ has an initial object the limit is evaluation in degree 0) constitute a Quillen equivalence
The canonical sSet-enrichment/tensoring/powering of the category of simplicial objects $[\Delta^{op}, C]$ makes $[\Delta^{op}, C]_{Reedy,S}$ (but not in general $[\Delta^{op}, C]_{proj,S}$) into a simplicial model category.
This is (Dugger, theorem 5.2, theorem 5.7, theorem 6.1).
So in particular every left proper combinatorial model category is Quillen equivalent to a simplicial model category.
We first show that the fibrant objects in $[\Delta^{op}, C]_{proj,S}$ are the objectwise fibrant objects all whose structure maps are weak equivalences in $C$. The argument for the fibrant objects in $[\Delta^{op}, C]_{Reedy,S}$ is directly analogous.
By general properties of left Bousfield localization, the fibrant objects in $[\Delta^{op}, C]_{proj,S}$ are the projective fibrant objects $X$ for which all induced morphisms on derived hom spaces
are weak equivalences. Since $s$ is cofibrant in $C$ by definition, also $s \cdot \Delta[k]$ is cofibrant in $[\Delta^{op}, C]_{proj}$.
So for $X \in [\Delta^{op}, C]_{proj}$ fibrant, let $X_\bullet \in [\Delta^{op}, [\Delta^{op}, C]]$ be a simplicial framing for it. Notice that this means that for all $n \in \mathbb{N}$ also $X_\bullet([n])$ is a simplicial framing for $X([n])$. This is because
$const X \to X_\bullet$ being a weak equivalence means that for all $n$ the morphism $X \to X_n$ is a weak equivalence, which means that for all $k$ the morphism $X([k]) \to X_n([k])$ is a weak equivalence.
$X_\bullet$ being fibrant in $[\Delta^{op}, [\Delta^{op}, C]_{proj}]_{Reedy}$ means that for all $n\in \mathbb{N}$ the morphism $X_{\Delta[n]} \to X_{\partial \Delta[n]}$ is a fibration in $[\Delta^{op}, C]_{proj}$, hence that for all $k \in \mathbb{N}$ the morphism $X_{\Delta[n]}([k]) \to X_{\partial \Delta[n]}([k])$ is a fibration in $C$, hence that $X([k])$ is Reedy fibrant.
Then we find
By assumption on the set $S$, this implies the claim.
$\,$
Now we show that the weak equivalences in $[\Delta^{op}, C]_{proj,S}$ are precisely those morphisms that become weak equivalences under the homotopy colimit.
By functorial cofibrant resolution and two-out-of-three, it is sufficient to show that this holds for morphisms between cofibrant objects.
By lemma 1, we have weak equivalences
seen by computing the derived homs by simplicial framings.
Now, by properties of left Bousfield localization, $A \to B$ is a weak equivalence if for all $S$-local objects $X$ the morphism $\mathbb{R}Hom(A \to B, X)$ is a weak equivalence. Looking at the diagram
we see that this is the case precisely if the vertical morphism on the right is a weak equivalence for all fibrant $Z \in C$, which is the case if $\lim_\to A \to \lim_\to B$ is a weak equivalence. Since $A$ and $B$ here are cofibrant in $[\Delta^{op}, C]_{proj}$, the colimits here are indeed homotopy colimits (as discussed there).
$\,$
Now we discuss that $(\lim_\to \dashv const): C \to [\Delta^{op}, C]_{proj,S}$ is a Quillen equivalence. First observe that on the global model structure $const : C \to [\Delta^{op}, C]_{proj}$ is clearly a right Quillen functor, hence we have a Quillen adjunction on the unlocalized structure. Moreover, by definition and by the above discussion, the derived functor of the left adjoint $\lim_\to$, namely the homotopy colimit, takes the localizing set $S$ to weak equivalences in $C$. Therefore the assumptions of the discussion at Quillen equivalence - Behaviour under localization are met, and hence it follows that $(\lim_\to \dashv const)$ descends as a Quillen adjunction also to the localization.
To see that this is a Quillen equivalence, it is sufficient to show that for $A \in [\Delta^{op}, C]_{proj}$ cofibrant and $Z \in C$ fibrant, a morphism $\lim_\to A \to Z$ is a weak equivalence in $C$ precisely if the adjunct $A \to const Z$ becomes a weak equivalence under the homotopy colimit.
For this notice that we have a commuting diagram
and so our statement follows (by 2-out-of-3) once we know that the vertical morphisms here are weak equivalences. The left one is because $A$ is cofibrant, by assumption, as before. To see that the right one is, too, consider the factorization
of the identity on $Z$, for any $i \in \mathbb{N}$. By lemma 1 the first morphism is a weak equivalence, and hence so is the morphism in question.
$\,$
Now we show that the weak equivalences in $[\Delta^{op}, C]_{Reedy,S}$ are the hocolim-equivalences.
By a general result on functoriality of localization, we have that the $(id \dashv id ) : [\Delta^{op}, C]_{Reedy,S} \stackrel{\leftarrow}{\to} [\Delta^{op}, C]_{proj,S}$ is at least a Quillen adjunction.
Let then $A \to B$ be a morphism in $[\Delta^{op}, C]$ and consider two fibrant replacements
where the first one ($\bar A \to \bar B$) is taken in $[\Delta^{op}, C]_{proj,S}$ and the second (\hat A \to \hat B) in $[\Delta^{op}, C]_{Reedy}$.
Assume first that $A \to B$ is a hocolim-equivalence. Then so is $\hat A \to \hat B$, because the horizontal morphisms are all objectwise weak equivalences. But $\hat A$ and $\hat B$ are fibrant in $[\Delta^{op}, C]_{Reedy}$, hence in $[\Delta^{op}, C]_{proj}$ by construction and at the same time all their structure maps are weak equivalences (use 2-out-of-3), so that they are in fact fibrant in $[\Delta^{op}, C]_{proj,S}$. By general properties of left Bousfield localization, weak equivalences between local fibrant objects are already weak equivalences in the unlocalized structure – so $\hat A \to \hat B$ is indeed even an objectwise weak equivalence. It follows then that so is $\bar A \to \bar B$, which is therefore in partiular a weak equivalence in $[\Delta^{op}, C]_{Reedy, S}$. Finally the left horizontal morphisms are also weak equivalences in $[\Delta^{op}, C]_{Reedy,S}$, by the above Quillen adjunction. So finally by 2-out-of-3 in $[\Delta^{op}, C]_{Reedy,S}$ it follows that also $A \to B$ is a weak equivalence there.
By an analogous diagram chase, one shows the converse implication holds, that $A \to B$ being a weak equivalence in $[\Delta^{op}, C]_{Reedy,S}$ implies that it is a hocolim-equivalence.
With this now it is clear that the identity adjunction above is in fact a Quillen equivalence.
$\,$
Finally we show that $(const \dashv ev_0) : [\Delta^{op}, C]_{Reedy,S} \stackrel{\overset{const}{\leftarrow}}{\underset{ev_0}{\to}} C$ is a Quillen equivalence.
First, it is immediate to check that $const : C \to [\Delta^{op}, C]_{Reedy}$ is left Quillen, and since $id : [\Delta^{op}, C]_{Reedy} \to [\Delta^{op}, C]_{Reedy,S}$ is left Quillen by definition of Bousfield localization, the above is at least a Quillen adjunction.
To see that it is a Quillen equivalence, let $A \in C$ be cofibrant and $X \in [\Delta^{op}, C]_{Reedy,S}$ be fibrant – which by the above means that it is a simplicial resolution – and consider a morphism $const A \to X$. We need to show that this is a weak equivalence, hence, by the above, that its hocolim is a weak equivalence, precisely if $A \to X_0$ is a weak equivalence in $C$.
To that end, find a cofibrant resolution $const \tilde A \to \tilde X$ of $const A \to X$ in $[\Delta^{op}, C]_{proj}$ and consider the diagram
The colimits on the right compute the homotopy colimit. By 2-out-of-3 it follows that the right vertical morphism is a weak equivalence precisely if the left vertical morphisms is.
$\,,$
Finally it remains to show that $[\Delta^{op}, C]_{Reedy,S}$ is a simplicially enriched model category. (…)
(uniqueness)
Let $C$ be a model category. Then there is a unique model category structure on $s C = [\Delta^{op}, C]$ such that
every morphism that is degreewise a weak equivalence in $C$ is a weak equivalence;
the cofibrations are those of the Reedy model structure;
the fibrant objects are the Reedy-fibrant objects whose face and degeneracy maps are weak equivalences in $C$.
This is (Rezk-Schwede-Shipley, theorem 3.1).
By theorem 1 at least one such model structure exists. By the discussion at model category – Redundancy of the axioms, the classes of cofibrations and fibrant objects already determine a model category structure.
For $C$ any left proper combinatorial model category, the derived hom-space between two objects $X, A$ may be computed by
choosing a cofibrant replacement $\hat X$ of $X$ in $C$;
choosing a Reedy fibrant replacement $\hat A$ of $const A$ in $[\Delta^{op}, C]$ such that all face and degeneracy maps are weak equivalences,
setting
By theorem 1 we may compute the derived hom space in $[\Delta^{op}, C]_{Reedy,S}$ after the inclusion $const : C \to [\Delta^{op}, C]$. Since by that theorem $[\Delta^{op}, C]_{Reedy,S}$ is a simplicial model category, by prop. 2 the derived hom space is given by the simplicial function complex between a cofibrant replacement of $const X$ and a fibrant replacement of $const A$. If $\hat X$ is cofibrant, then $const \hat X$ is already Reedy cofibrant, and by the theorem $\hat A$ as stated is a a fibrant resolution of $const A$. Finally, the theorem says that the simplicial function complex is given by
There is also a version for stable model categories:
Every proper cofibrantly generated stable model category is Quillen equivalent to a simplicial model category
This is (Rezk-Schwede-Shipley, prop 1.3).
A particularly important type of simplicial model categories are those that are also combinatorial model categories.
A combinatorial simplicial model category is precisely a presentation for a locally presentable (∞,1)-category. See there for more details.
The definition appears in
Textbook references include
Philip Hirschhorn, section 9.1.5 of Model categories and their localizations, volume 99 of Mathematical Surveys and Monographs , American Mathematical Society, 2009.
Jacob Lurie, section A.3 in Higher Topos Theory
Further review includes
Further developments include
Charles Rezk, Stefan Schwede, Brooke Shipley, Simplicial structures on model categories and functors, American Journal of Mathematics, Vol. 123, No. 3 (Jun., 2001), pp. 551-575 (arXiv:0101162, jstor)
Dan Dugger, Replacing model categories with simplicial ones, Trans. Amer. Math. Soc. vol. 353, number 12 (2001), 5003-5027. (pdf)