nLab
proper model category

Idea

In a model category fibrations enjoy pullback stability and cofibrations are stable under pushout, but weak equivalences need not have either property. In a proper model category weak equivalences are also preserved under certain pullbacks and/or certain pushouts.

Definition

A model category is called

• right proper if weak equivalences are preserved by pullback along fibrations

• left proper if weak equivalence are preserved by pushout along cofibrations

• proper if it is both left and right proper.

So a model category is right proper if for every weak equivalence $f:A\to B$ in $W\subset \mathrm{Mor}(C)$ and every fibration $h:C\to B$ the pullback $h^{*}f:A{\times }_{B}C\to C$ in

$$\begin{array}{ccc}A{\times }_{C}B& \to & A\\ {\downarrow }^{\Rightarrow h^{*}f\in W}& & {\downarrow }^{f\in W}\\ C& \stackrel{h\in F}{\to }& B\end{array}$$\array{ A \times_C B &\to& A \\ \;\;\downarrow^{\mathrlap{\Rightarrow h^* f \in W}} && \downarrow^{\mathrlap{f \in W}} \\ C &\stackrel{h \in F}{\to}& B }

is a weak equivalence.

Examples

Left proper model categories

• every model category in which all objects are cofibrant is left proper;

  this includes notably

  • the standard model structure on simplicial sets

  • the model structure for quasi-categories

  • the model structure on simplicially enriched categories

  and many model structures derived from these, such as

  • the injective global model structure on simplicial presheaves over any (simplicial) category;

• the left Bousfield localization of every left proper combinatorial model category at a set of morphisms is again left proper.

  So in particular also the local injective model structures on simplicial presheaves over a site are left proper.

Non-left proper model categories

A class of model structures which tends to be not left proper are model structures on categories of not-necessarily commutative algebras.

For instance

• the standard model structure on simplicial associative unital algebras (weak equivalences and fibrations are those of the underlying simplicial sets) is not left proper (see Rezk2000, example 2.11)

But it is Quillen equivalent to a model structure that is left proper. This is discussed below.

Right proper model categories

• every model category in which each object is fibrant is right proper.

  This includes for instance the standard model structure on topological spaces.

Proper model categories

Model categories which are both left and right proper include

• Top: model structure on topological spaces

• sSet: model structure on simplicial sets

• The global model structure on simplicial presheaves and any local such model structure over a site with topos point? and weak equivalences the stalkwise weak equivalences.

• The standard model structures on chain complexes.

Proper Quillen equivalent model structures

While some model categories fail to be proper, often there is a Quillen equivalent one that does enjoy properness.

Theorem

Every model category whose acyclic cofibrations are monomorphisms is Quillen equivalent to its model structure on algebraic fibrant objects. In this all objects are fibrant, so that it is right proper.

Proof

This is theorem 2.18 in

• Thomas Nikolaus, Algebraic models for higher categories (arXiv)

Theorem

Let $T$ be a simplicial (possibly multi-colored) theory, and let $T\mathrm{Alg}$ be the corresponding category of simplicial T-algebras. This carries a model category structure where the fibrations and weak equivalences are those of the underlying simplicial sets in the standard model structure on simplicial sets.

Then there exists a morphism of simplicial theories $T\to S$ such that

1. the induced adjunction $S\mathrm{Alg}\overrightarrow{\leftarrow }T\mathrm{Alg}$ is a Quillen equivalence;

2. $S\mathrm{Alg}$ is a proper simplicial model category.

Proof

This is the content of

• Charles Rezk, Every homotopy theory of simplicial algebras admits a proper model (math/0003065)

Properties

Homotopy (co)limits in proper model categories

Lemma

In a left proper model category, ordinary pushouts along cofibrations are homotopy pushouts.

Dually, in a right proper model category, ordinary pullbacks along fibrations are homotopy pullbacks.

Proof

This is stated for instance in HTT, prop A.2.4.4 or in prop. 1.19 in Bar. We follow the proof given in this latter reference.

We demonstrate the first statement, the second is its direct formal dual.

So consider a pushout diagram

$$\begin{array}{ccc}K& \to & Y\\ {\downarrow }^{\in \mathrm{cof}}& & \downarrow \\ L& \to & X\end{array}$$\array{ K &\to& Y \\ \downarrow^{\mathrlap{\in cof}} && \downarrow \\ L &\to& X }

in a left proper model category, where the morphism $K\to L$ is a cofibration, as indicated. We need to exhibit a weak equivalence $X\prime \stackrel{}{\to }X$ from an object $X\prime$ that is manifestly a homotopy pushout of $L\leftarrow K\to Y$.

The standard procedure to produce this $X\prime$ is to pass to a weakly equivalent diagram with the property that all objects are cofibrant and one of the morphisms is a cofibration. The ordinary pushout of that diagram is well known to be the homotopy pushout, as described there.

So pick a cofibrant replacement $\varnothing \hookrightarrow K\prime \stackrel{\simeq }{\to }$ of $K$ and factor $K\prime \to K\to Y$ as a cofibration followed by a weak equivalence $K\prime \hookrightarrow Y\prime \stackrel{\simeq }{\to }Y$ and similarly factor $K\prime \to K\to L$ as $K\prime \hookrightarrow L\prime \stackrel{\simeq }{\to }L$

This yields a weak equivalence of diagrams

$$\begin{array}{ccc}Y& \stackrel{\simeq }{\leftarrow }& Y\prime \\ \uparrow & & {\uparrow }^{\in \mathrm{cof}}\\ K& \stackrel{\simeq }{\leftarrow }& K\prime \\ {\downarrow }^{\in \mathrm{cof}}& & {\downarrow }^{\in \mathrm{cof}}\\ L& \stackrel{\simeq }{\leftarrow }& L\prime \end{array},$$\array{ Y &\stackrel{\simeq}{\leftarrow}& Y' \\ \uparrow && \uparrow^{\mathrlap{\in cof}} \\ K &\stackrel{\simeq}{\leftarrow}& K' \\ \downarrow^{\mathrlap{\in cof}} && \downarrow^{\mathrlap{\in cof}} \\ L &\stackrel{\simeq}{\leftarrow}& L' } \,,

where now the diagram on the right is cofibrant as a diagram, so that its ordinary pushout

$$X\prime :=L\prime \coprod _{K\prime }Y\prime$$X' := L' \coprod_{K'} Y'

is a homotopy colimit of the original diagram. To obtain the weak equivalence from there to $X$, first form the further pushouts

$$\begin{array}{ccccccc}K& & & \to & & & Y\\ & {\nwarrow }^{\in W}& & & & {\nearrow }_{\simeq }& \\ & & K\prime & \to & Y\prime & & \\ {\downarrow }^{\in \mathrm{cof}}& & {\downarrow }^{\in \mathrm{cof}}& & {\downarrow }^{\in \mathrm{cof}}& & \downarrow \\ & & L\prime & \to & X\prime & & \\ & {}^{\in W}\swarrow & & & & {\searrow }^{\simeq }& \\ L″:=K\coprod _{K\prime }L& & & \to & & & L″\coprod _{K}Y\\ {\downarrow }^{\in W}& & & & & & \downarrow \\ L& & & \to & & & X\end{array},$$\array{ K &&&\to&&& Y \\ & \nwarrow^{\mathrlap{\in W}} &&&& \nearrow_{\mathrlap{\simeq}} & \\ && K' &\to& Y' && \\ \downarrow^{\mathrlap{\in cof}} && \downarrow^{\mathrlap{\in cof}} && \downarrow^{\mathrlap{\in cof}} && \downarrow \\ && L' &\to& X' && \\ & {}^{\mathllap{\in W}} \swarrow &&&& \searrow^{\mathrlap{\simeq}} & \\ L'':= K \coprod_{K'} L &&&\to&&& L'' \coprod_{K} Y \\ \downarrow^{\mathrlap{\in W}} &&&&&& \downarrow \\ L &&&\to&&& X } \,,

where the total outer diagram is the original pushout diagram. Here the cofibrations are as indicated by the above factorization and by their stability under pushouts, and the weak equivalences are as indicated by the above factorization and by the left properness of the model category. The weak equivalence $L″\stackrel{\simeq }{\to }L$ is by the 2-out-of-3 property.

This establishes in particular a weak equivalence

$$X\prime \stackrel{\simeq }{\to }L″\coprod _{K}Y.$$X' \stackrel{\simeq}{\to} L'' \coprod_K Y \,.

It remains to get a weak equivalence further to $X$. For that, take the two outer squares from the above

$$\begin{array}{ccc}K& \to & Y\\ {\downarrow }^{\in \mathrm{cof}}& & \downarrow \\ L″& \to & L″\coprod _{K\prime }Y\\ {\downarrow }^{\in W}& & \downarrow \\ L& \to & X\end{array}.$$\array{ K &\to& Y \\ \downarrow^{\mathrlap{\in cof}} && \downarrow \\ L'' &\to& L'' \coprod_{K'} Y \\ \downarrow^{\mathrlap{\in W}} && \downarrow \\ L &\to& X } \,.

Notice that the top square is a pushout by construction, and the total one by assumption. Therefore by the general theorem about pastings of pushouts, also the lower square is a pushout.

Then factor $K\to Y$ as a cofibration followed by a weak equivalence $K\hookrightarrow Z\stackrel{\simeq }{\to }Y$ and push that factorization through the double diagram, to obtain

$$\begin{array}{ccccc}K& \stackrel{\in \mathrm{cof}}{\to }& Z& \stackrel{\in W}{\to }& Y\\ {\downarrow }^{\in cof}& & {\downarrow }^{\in \mathrm{cof}}& & \downarrow \\ L″& \stackrel{\in \mathrm{cof}}{\to }& L″\coprod _{K}Z& \stackrel{\in W}{\to }& L″\coprod _{K\prime }Y\\ {\downarrow }^{\in W}& & {\downarrow }^{\in W}& & \downarrow \\ L& \to & L\coprod _{K}Z& \stackrel{\in W}{\to }& X\end{array}.$$\array{ K &\stackrel{\in cof}{\to}& Z &\stackrel{\in W}{\to}& Y \\ \downarrow^{\mathrlap{\in \cof}} && \downarrow^{\mathrlap{\in cof}} && \downarrow \\ L'' &\stackrel{\in cof}{\to}& L'' \coprod_{K} Z &\stackrel{\in W}{\to}& L'' \coprod_{K'} Y \\ \downarrow^{\mathrlap{\in W}} && \downarrow^{\mathrlap{\in W}} && \downarrow \\ L & \to& L \coprod_K Z &\stackrel{\in W}{\to}& X } \,.

Again by the behaviour of pushouts under pasting, every single square and composite rectangle in this diagram is a pushout. Using this, the cofibration and weak equivalence properties from before push through the diagram as indicated. This finally yields the desired weak equivalence

$$L″\coprod _{K\prime }Y\stackrel{\simeq }{\to }X$$L'' \coprod_{K'} Y \stackrel{\simeq}{\to} X

by 2-out-of-3.

If we had allowed ourselved to assume in addition that $K$ itself is already cofibrant, then the above statement has a much simpler proof, which we list just for fun, too.

Proof of the above assuming that the domain of the cofibration is cofibrant

Let $A\hookrightarrow B$ be a cofibration with $A$ cofibrant and let $A\to C$ be any other morphism. Factor this morphism as $A\hookrightarrow C\prime \stackrel{\simeq }{\to }C$ by a cofibration followed by an acyclic fibration. This give a weak equivalence of pushout diagrams

$$\begin{array}{ccc}C\prime & \stackrel{\simeq }{\to }& C\\ \uparrow & & \uparrow \\ A& \stackrel{=}{\to }& A\\ \downarrow & & \downarrow \\ B& \stackrel{=}{\to }& B\end{array}.$$\array{ C' &\stackrel{\simeq}{\to}& C \\ \uparrow && \uparrow \\ A &\stackrel{=}{\to}& A \\ \downarrow && \downarrow \\ B &\stackrel{=}{\to}& B } \,.

In the diagram on the left all objects are cofibrant and one morphism is a cofibration, hence this is a cofibrant diagram and its ordinary colimit is the homotopy colimit. Using that pushout diagrams compose to pushout diagrams, that cofibrations are preserved under pushout and that in a left proper model category weak equivalences are preserved under pushout along cofibrations, we find a weak equiovalence $\mathrm{hocolim}\stackrel{\simeq }{\to }B{\coprod }_{A}C$

$$\begin{array}{ccccc}A& \stackrel{\in \mathrm{cof}}{\to }& C\prime & \stackrel{\in W\cap \mathrm{fib}}{\to }& C\\ {\downarrow }^{\in \mathrm{cof}}& & {\downarrow }^{\in \mathrm{cof}}& & {\downarrow }^{\in \mathrm{cof}}\\ B& \to & \mathrm{hocolim}& \stackrel{\in W}{\to }& B\coprod _{A}C\end{array}.$$\array{ A &\stackrel{\in cof}{\to}& C' &\stackrel{\in W \cap fib}{\to}& C \\ \downarrow^{\mathrlap{\in cof}} && \downarrow^{\mathrlap{\in cof}} && \downarrow^{\mathrlap{\in cof}} \\ B &\to& hocolim &\stackrel{\in W}{\to}& B \coprod_A C } \,.

The proof for the second statement is the precise formal dual.

References

The usefulness of right properness for constructions of homotopy categories is discussed in

• J. Jardine, Cocycle categories (pdf)

The general theory can be found in Chapter 13 of

• P. Hirschhorn, Model categories and their localizations, volume 99 of Mathematical Surveys and Monographs , American Mathematical Society, 2009.