nLab
combinatorial model category

Idea

A combinatorial model category is a particularly tractable model category structure. (Notice however that there is also the, closely related, technical notion of a tractable model category).

Being combinatorial means that there is very strong control over the cofibrations in these model structures: there is a set (meaning small set, not proper class) of generating (acyclic) cofibrations, and all objects, in particular the domains and codomains of these cofibrations, are small objects.

So as a slogan we have that

A combinatorial model structure is one that is generated from small data : it is generated from a small set of (acyclic) cofibrations between small objects.

In fact, the combinatoriality condition is a bit stronger than that, as it requires even that every object is small and is the colimit over a small set of generating objects. If one instead just has that the domains of the generating cofibrations are small objects, then we have a cellular model category.

There exist large classes of model categories that either are combinatorial or, if not, are Quillen equivalent to ones that are. See the list of examples below.

The relevance of combinatorial model categories is given more abstractly by the result that

Combinatorial simplicial model categories are precisely those model categories that model presentable (∞,1)-categories.

Definition

Definition (Jeff Smith)

A model category $C$ is combinatorial if it is

• locally presentable

and

• cofibrantly generated.

Recall from the discussion at cofibrantly generated model category that this means that $C$ has a set (not a proper class) $I$ of generating cofibrations and and a set of trivial generating cofibrations in that

$$\mathrm{cof}=\mathrm{llp}(\mathrm{rlp}(I))$$cof = llp(rlp(I))

$$\mathrm{fib}=\mathrm{llp}(\mathrm{rlp}(J)).$$fib = llp(rlp(J)) \,.

Here $\mathrm{fib},\mathrm{cof}\subset \mathrm{Mor}(C)$ is the collection of fibrations and cofibration, respectively, and $\mathrm{llp}(S),\mathrm{rlp}(S)$ is the collection of morphisms satisfying the left or right, respectively, lifting property with respect to a collection of morphisms $S$.

Jeff Smith’s theorem, below, gives an equivalent characterization that is usually easier to handle.

Characterization theorems

There are two powerful theorems that characterize combinatorial model categories in terms of data that is often easier to handle:

• Jeff Smith’s theorem characterizes combinatorial model categories just in terms of weak equivalences and generating cofibrations, hence using ony two third of the input data explicitly required. This greatly facilitates identifying combinatorial model category structures. For instance it helps to show that the left Bousfield localization of certain combinatorial model categories is again a combinatorial model category.

• Dan Dugger’s theorem shows that combinatorial simplicial model categories are, up to Quillen equivalence, precisely those model categories that have a presentation in that they are Bousfield localizations of global global model structures on simplicial presheaves. This was used by Jacob Lurie to show that simplicial combinatorial model categories are precisely the models for locally presentable (∞,1)-categories.

Smith’s theorem

A central theorem about combinatorial model categories is Jeff Smith’s theorem which establishes the existence of combinatorial model category structures from a small amount of input data.

Theorem (Jeff Smith’s theorem)

For

• $C$ a locally presentable category;

• ${\mathrm{Arr}}_W(C)\subset \mathrm{Arr}(C)$ an accessibly embedded accessible full subcategory of the arrow category $\mathrm{Arr}(C)$ on a class $W\subset \mathrm{Mor}(C)$;

• $I\subset \mathrm{Mor}(C)$ a small set (not a proper class) of morphisms of $C$

such that

• $W$ satisfies 2-out-of-3;

• $\mathrm{inj}(I)\subset W$

• $\mathrm{cof}(I)\cap W$ is closed under pushout and transfinite composition.

we have that $C$ is a combinatorial model category with

• weak equivalences $W$;

• cofibrations $\mathrm{cof}(I)$.

Moreover, every combinatorial model category arises in this way.

Here the notation is as described at cofibrantly generated model category, so: $\mathrm{inj}(I)=\mathrm{rlp}(I)$ and $\mathrm{cof}(I)=\mathrm{llp}(\mathrm{rlp}(I))$.

This statement was announced by Jeff Smith in 1998 at a conference in Barcelona and appararently first appeared in print as theorem 1.7 in

• Tibor Beke, Sheafifiable homotopy model categories (arXiv)

The above formulation follows prop 2.2 in

• Clark Barwick, On left and right model categories and left and right Bousfield localization (pdf)

Proof

To prove the first part of the statement, that the given data encodes a combinatorial model category, it is sufficient to find a small set $J$ such that

$$\mathrm{cof}(J)=W\cap \mathrm{cof}(I).$$cof(J) = W \cap cof(I) \,.

With that the statement follows using the small object argument to show the existence of the required factorizations.

To find this small set, we make use of the assumption that the subcategory ${\mathrm{Arr}}_W(C)\subset \mathrm{Arr}(C)$ of weak equivalences and commuting squares in $C$ between them is an accessible subcategory of the arrow category $\mathrm{Arr}(C)$. This means that there is a small set $W_0\subset W$ such that every element of $W$ is a $\kappa$-directed colimit over element in $W_0$ in ${\mathrm{Arr}}_W(C)$, for some cardinal number $\kappa$. We set then

$$J:=W_0\cap \mathrm{cof}(I).$$J := W_0 \cap cof(I) \,.

By the assumption that ${\mathrm{Arr}}_C(W)$ is an accessibly embedded accessible subcategory of $\mathrm{Arr}(C)$ and that $C$ itself is locally presentable, so that all its objects are small objects, it follows that every morphism

$$\begin{array}{ccc}K& \to & M\\ \downarrow & & {\downarrow }^{\in W}\\ L& \to & N\end{array}$$\array{ K &\to& M \\ \downarrow && \downarrow^{\mathrlap{\in W}} \\ L &\to& N }

in $\mathrm{Arr}(C)$ lifts through one of the components in $W_0$ of $W$ (this mechanism is described in detail at small object) as

$$\begin{array}{ccccc}K& \to & P& \to & M\\ \downarrow & & {\downarrow }^{\in W_0}& & {\downarrow }^{\in W}\\ L& \to & Q& \to & N\end{array}.$$\array{ K &\to& P &\to& M \\ \downarrow && \downarrow^{\mathrlap{\in W_0}} && \downarrow^{\mathrlap{\in W}} \\ L &\to& Q &\to& N } \,.

In the special case that $K\to L$ is in $I$, we can refine this to a factoring through $J=W\cap \mathrm{cof}(I)$ as follows:

using the small object argument factor the canonical morphism $L{\coprod }_{K}P\to Q$ as $L{\coprod }_{K}P\stackrel{\in \mathrm{cell}(I)}{\to }R\stackrel{\in \mathrm{inj}(J)}{\to }Q$. Then $L\to Q$ lifts to $L\to R$ and we obtain the factorization

$$\begin{array}{ccccc}K& \to & P& \to & M\\ {\downarrow }^{\in I}& & {\downarrow }^{\in W_0\cap \mathrm{cof}(I)}& & {\downarrow }^{\in W}\\ L& \to & R& \to & N\end{array}$$\array{ K &\to& P &\to& M \\ \downarrow^{\mathrlap{\in I}} && \downarrow^{\mathrlap{\in W_0 \cap cof(I)}} && \downarrow^{\mathrlap{\in W}} \\ L &\to& R &\to& N }

of the original square from an element in $I$ to an element in $W$ through an element in $J=W\cap \mathrm{cof}(I)$. (In Beke, following Jeff Smith, this is called the solution set condition: $W_0\cap \mathrm{cof}(I)$ is “a solution set for $W$ at $I$”).

By following now precisely the small object argument with the only difference that one factors all the squares over which one takes a colimit in that argument through elements in $J$ as above, it follows now that every morphism $A\stackrel{\in W}{\to }B$ in $W$ may be factored as

$$A\stackrel{\in \mathrm{cell}(J)}{\to }C\stackrel{\in \mathrm{inj}(I)}{\to }B.$$A \stackrel{\in cell(J)}{\to} C \stackrel{\in inj(I)}{\to} B \,.

This we use to show that every morphism $f\in \mathrm{cof}(I)\cap W$ is in $\mathrm{cof}(J)$:

since $f\in W$ we may factor $f$ as above and since $f\in \mathrm{cof}(I)$ we obtain a lift $\sigma$ in

$$\begin{array}{ccc}A& \stackrel{\in \mathrm{cell}(J)}{\to }& C\\ {\downarrow }^f& {}^{\sigma }\nearrow & {\downarrow }^{\in \mathrm{inj}(I)}\\ B& \stackrel{=}{\to }& B\end{array}.$$\array{ A &\stackrel{\in cell(J)}{\to}& C \\ \downarrow^{f} &{}^\sigma \nearrow& \downarrow^{\mathrlap{\in inj(I)}} \\ B &\stackrel{=}{\to}& B } \,.

Rearranging this it becomes a retract diagram in $\mathrm{Arr}(C)$

$$\begin{array}{ccccc}A& \stackrel{=}{\to }& A& \stackrel{=}{\to }& A\\ {\downarrow }^f& & {\downarrow }^{\in \mathrm{cell}(J)}& & {\downarrow }^f\\ B& \stackrel{\sigma }{\to }& C& \stackrel{\in \mathrm{inj}(I)}{\to }& B\end{array}$$\array{ A &\stackrel{=}{\to}& A &\stackrel{=}{\to}& A \\ \downarrow^f && \downarrow^{\mathrlap{\in cell(J)}} && \downarrow^f \\ B &\stackrel{\sigma}{\to}& C & \stackrel{\in inj(I)}{\to} & B }

which shows that $f$ is a retract of an element in $\mathrm{cell}(J)\subset \mathrm{cof}(J)$, hence itself in $\mathrm{cof}(J)$.

And the converse statement is immediate: by definition $J\subset \mathrm{cof}(I)\cap W$ and $\mathrm{cof}(J)$ is the saturation of $J$ under the operation of forming retracts of transfinite compositions of pushouts of elements of $J$, under which $\mathrm{cof}(I)\cap W$ is assumed to be closed.

In total we have indeed $\mathrm{cof}(J)=\mathrm{cof}(I)\cap W$ which shows that the $I$ and $W$ given to determine a combinatorial model category.

To see the converse, that every combinatorial model structure arises this way, it is sufficient to show that for every combinatorial model category the category ${\mathrm{Arr}}_W(C)\subset \mathrm{Arr}(C)$ is an accessible category.

Dugger’s theorem

Theorem (Dan Dugger)

Every combinatorial model category $C$ is Quillen equivalent to a left Bousfield localization $L_S\mathrm{SPSh}(K{)}_{\mathrm{proj}}$ of the global projective model structure on simplicial presheaves $\mathrm{SPSh}(K{)}_{\mathrm{proj}}$ on a small category $K$

$$L_S\mathrm{SPSh}(K{)}_{\mathrm{proj}}\stackrel{{\simeq }_{\mathrm{Quillen}}}{\to }C.$$L_S SPSh(K)_{proj} \stackrel{\simeq_{Quillen}}{\to} C \,.

Proof

This is theorem 1.1. in Dug00.

Notice that the theorem just mentions plain combinatorial model categories, not simplicial model categories. But of course by basic facts of enriched category theory $\mathrm{Funct}(C^{\mathrm{op}},\mathrm{SSet})$ is an SSet-enriched category and its projective global model structure on functors $\mathrm{Func}(C^{\mathrm{op}},\mathrm{SSet}{)}_{\mathrm{proj}}$ is compatibly a simplicial model category, as are all its Bousfield localizations. (See model structure on simplicial presheaves for more details.) Therefore an immediate but very useful corollary of the above statement is

Corollary

Every combinatorial model category is Quillen equivalent to one which is

• a simplicial model category

• a left proper model category.

Tractable combinatorial model categories

A combinatorial model category is a tractable model category precisely if the set $I$ of generating cofibrations can be chosen such that all elements have a cofibrant object as domain.

A left proper combinatorial model category precisely if the set $J$ of generating trivial cofibrations can be chosen with cofibrant domain.

This are corollaries 2.7 and 2..8 in Bar.

Properties

Homotopy colimits

Proposition

In a combinatorial model category, for every sufficiently large regular cardinal $\kappa$ the following holds:

• $\kappa$-filtered colimits preserve weak equivalences;

• hence $\kappa$-filtered colimits are already homotopy colimits.

Proof

This appears as proposition 7.3 in Dug00, reproduced for instance as prop. 2.5 in Bar.

The point is to choose $\kappa$ such that all domains and codomains of the generating cofibrations are $\kappa$-compact object. This is possible since by assumption that $C$ is a locally presentable category all its objects are small objects, hence each a $\lambda$-compact object for some cardinal $\lambda$. Take $\kappa$ to be the maximum of these.

Let $F,G:J\to C$ be $\kappa$-filtered diagrams in $C$ and $F\to G$ a natural transformation that is degreewise a weak equivalence. Using the functorial factorization provided by the small object argument this may be factored as $F\to H\to G$ where the first transformation is objectwise an acyclic cofibration and the second objectwise an acyclic fibration, and by functoriality of the factorization this sits over a factorization

$$\underrightarrow{\lim }F\stackrel{\simeq }{\hookrightarrow }\underrightarrow{\lim }H\stackrel{}{\to }\underrightarrow{\lim }G.$$\lim_\to F \stackrel{\simeq}{\hookrightarrow} \lim_\to H \stackrel{}{\to}\lim_\to G \,.

It remains to show that the second morphism is a weak equivalence. But by our factroization and by 2-out-of-3 applied to our comoponentwise weak equivalences, we have that all its components $H(j)\to G(j)$ are acyclic fibrations .

At small object is is described in detail how $\kappa$-smallness of an object $X$ implies that morphisms from $X$ into a $\kappa$-filtered colimit lift to some component of the colimit

$$\begin{array}{ccccccccc}\cdots & \to & H(j-1)& \to & H(j)& \to & H(j+1)& \to & \cdots \\ & & & {}^{\exists \hat{f}}\nearrow & \downarrow & \swarrow \\ & & X& \stackrel{\forall f}{\to }& \underrightarrow{\lim }H\end{array}.$$\array{ \cdots&\to&H(j-1) &\to& H(j) &\to& H(j+1) &\to& \cdots \\ &&&{}^{\mathllap{\exists \hat f}}\nearrow&\downarrow & \swarrow \\ &&X& \stackrel{\forall f}{\to} &\lim_\to H } \,.

So given a diagram

$$\begin{array}{ccc}X& \to & \underrightarrow{\lim }H\\ {\downarrow }^{\in I}& & \downarrow \\ Y& \to & \underrightarrow{\lim }G\end{array}$$\array{ X &\to& \lim_\to H \\ \downarrow^{\mathrlap{\in I}} && \downarrow \\ Y &\to& \lim_\to G }

we are guaranteed, by the $\kappa$-smallness of $X$ and $Y$ that we established above, a lift

$$\begin{array}{ccccc}X& \to & H(j)& \to & \underrightarrow{\lim }H\\ {\downarrow }^{\in I}& & {\downarrow }^{\in \in \mathrm{rlp}(I)}& & \downarrow \\ Y& \to & G(j)& \to & \underrightarrow{\lim }G\end{array}$$\array{ X &\to& H(j) &\to& \lim_\to H \\ \downarrow^{\mathrlap{\in I}} && \downarrow^{\in \mathrlap{\in rlp(I)}} && \downarrow \\ Y &\to& G(j) &\to& \lim_\to G }

into some component at $j\in J$ and hence a lift

$$\begin{array}{ccccc}X& \to & H(j)& \to & \underrightarrow{\lim }H\\ {\downarrow }^{\in I}& \nearrow & {\downarrow }^{\in \in \mathrm{rlp}(I)}& & \downarrow \\ Y& \to & G(j)& \to & \underrightarrow{\lim }G\end{array}.$$\array{ X &\to& H(j) &\to& \lim_\to H \\ \downarrow^{\mathrlap{\in I}} & \nearrow & \downarrow^{\in \mathrlap{\in rlp(I)}} && \downarrow \\ Y &\to& G(j) &\to& \lim_\to G } \,.

Thereby ${\lim }_{\to }H\to {\lim }_{\to }G$ is in $\mathrm{rlp}(I)\subset W$.

Bousfield localization

Combinatorial model categories, lile cellular model categories have a good theory of Bousfield localizations, at least if in addition they are left proper. See Bousfield localization of model categories for more on this.

Examples

Basic examples

Basic examples are

• SSet both with the standard and with the Joyal-model structure on simplicial sets

• the category of dendroidal sets with its model structure on dendroidal sets

• the categories of $(n,r)$-Theta spaces.

Derived examples

Further classes of examples are obtained from such basic examples by localizing presheaf categories with values in these:

• For $V$ a combinatorial model category and $C$ a small category the injective and projective global model structure on functors $\mathrm{Funct}(C,V{)}_{\mathrm{inj}}$ and $\mathrm{Funct}(C,V{)}_{\mathrm{proj}}$ are again combinatorial model categories. (e.g. HTT, prop. A.2.8.2)

• If $V$ is a left or right proper model category then so is $\mathrm{Funct}(C,V{)}_{\mathrm{inj}}$ and $\mathrm{Funct}(C,V{)}_{\mathrm{proj}}$ and hence the standard results of the theory of Bousfield localization of model categories applies, which ensures that all Bousfield localizations $L_S\mathrm{Funt}(C,V)$ are again combinatorial model categories. Such local local model structures on homotopical presheaves includes notably the local model structure on simplicial presheaves.

From cofibrantly generated model categories

Not every cofibrantly generated model category is also a combinatorial model category.

For instance:

(Counter)example

Top with the standard model structure on topological spaces is cofibrantly generated, but not combinatorial. But it is Quillen equivalent to a combinatorial model structure, namely to the standard model structure on simplicial sets (see homotopy hypothesis).

One might therefore ask which cofibrantly generated model categories are Quillen equivalent to combinatorial ones. It turns out that if one assumes the large-cardinal hypothesis Vopěnka's principle?, then every cofibrantly generated model category is Quillen equivalent to a combinatorial one. In fact, if we slightly generalize the notion of “cofibrantly generated,” this statement is equivalent to Vopěnka’s principle. For a discussion of this see

• J. Rosicky, Are all cofibrantly generated model categories combinatorial? (ps)

Although Vopěnka’s principle cannot be proven from ZFC, and in fact is fairly strong as large cardinal hypotheses go, this means that looking for cofibrantly generated model categories that are not Quillen equivalent to combinatorial ones is probably a waste of time. Certainly, all known cofibrantly generated model categories are Quillen equivalent to simplicial ones, usually in a fairly natural way.

Simplicial combinatorial model categories

Those combinatorial model categories that are at the same time simplicial model categories are precisely those that present presentable (∞,1)-categories. See combinatorial simplicial model category.

References

Much of the theory of combinatorial model categories goes back to Jeff Smith. Apparently Smith will eventually present a book on this subject. To date, however, his ideas and results appear reproduced in articles of other authors.

After Smith presented his recognition theorem at a conference in Barcelona, its first appearance in a publication is apparently lemma 1.8 in

• Tibor Beke, Sheafifiable homotopy model categories (arXiv)

The very definition of combinatorial model categories appears also for instance as definition A.2.6.1 in

• Jacob Lurie, Higher Topos Theory;

or definition 1.3 in

• Clark Barwick, On (enriched) left Bousfield localization of model categories (arXiv)

Smith’s theorem appears as proposition A.2.6.8 in HTT, as proposition 2.2 in Bar.

Dugger’s theorem is in

• Daniel Dugger, Combinatorial model categories have presentations (arXiv)