Contents

for ∞-groupoids

# Contents

## 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.

Put differently, in a proper model category, homotopy pullbacks and/or pushouts can be computed with less need for fibrant and/or cofibrant replacement.

## Definition

###### Definition

A model category is called

###### Remark

More in detail this means the following. A model category is right proper if for every weak equivalence $f : A \to B$ in $W\subset Mor(C)$ and every fibration $h : C \to B$ the pullback $h^* f : A \times_B C \to C$ in

$\array{ A \times_C B &\longrightarrow& A \\ \;\;\big\downarrow{}^{\mathrlap{\Rightarrow h^* f \in W}} && \big\downarrow{}^{\mathrlap{f \in W}} \\ C &\underset{h \in F}{\longrightarrow}& B }$

is a weak equivalence.

###### Remark

The above definition is the way it is usually phrased, but in fact it is equivalent to a seemingly weaker condition that is sometimes easier to check: for right properness it suffices to assume that weak equivalences are preserved by pullback along fibrations between fibrant objects. That is, in the more explicit version above, we are free to assume that $B$ (hence also $C$) is fibrant; this then implies the more general version without this hypothesis. See Proposition below.

## Examples

### Left proper model categories

• by cor. , every model category in which all objects are cofibrant is left proper;

this includes notably

and many model structures derived from these, such as

• 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

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

### Proper model categories

Model categories which are both left and right proper include

### 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.

###### Theorem

Let $T$ be a simplicial (possibly multi-colored) theory, and let $T Alg$ 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 classical model structure on simplicial sets.

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

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

2. $S Alg$ is a proper simplicial model category.

This is the content of (Rezk 02)

## Properties

### General

The following says that left/right properness holds locally in every model category, namely between cofibrant/fibrant objects.

###### Proposition

Given a model category,

1. every pushout of a weak equivalence between cofibrant objects along a cofibration is again a weak equivalence;

2. every pullback of a weak equivalence between fibrant objects along a fibration is again a weak equivalence.

A proof is spelled out in (Hirschhorn, prop. 13.1.2), there attributed to (Reedy).

This gives a large class of examples of left/right proper model categories:

###### Corollary

A model category in which all objects are cofibrant is left proper.

A model category in which all objects are fibrant is right proper.

See in the list of Examples below for concrete examples.

Notice that the prop. applies only (in the right proper case, for concreteness) to pullbacks of fibrations along weak equivalences in which all three objects are fibrant, since a fibration with fibrant codomain also has fibrant domain. The definition of right proper, on the other hand, states this property in the case when none of the objects are assumed to be fibrant.

One might consider as an “in-between” assumption the situation when only the common codomain of the fibration and the weak equivalence (hence also the domain of the fibration) are fibrant; but it turns out that this apparently-weaker assumption is sufficient to imply full right properness. This can be found, for instance, as Lemma 9.4 of Bousfield 2001.

###### Proposition

Suppose that in some model category, if $X\to Y$ is a fibration and $Z\to Y$ a weak equivalence, with $Y$ (hence also $X$) fibrant, then the pullback $X\times_Y Z \to X$ is a weak equivalence. Then the model category is right proper, i.e. the same statement is true without the assumption that $Y$ is fibrant.

###### Proof

Suppose given $X\to Y\leftarrow Z$ where $X\to Y$ is a fibration and $Z\to Y$ a weak equivalence. Choose a fibrant replacement $Y\to R Y$, and factor $X\to Y \to R Y$ as a weak equivalence $X\to R X$ followed by a fibration $R X \to R Y$. The assumption now applies to the cospan $R X \to R Y \leftarrow Y$, so that the map $R X \times_{R Y} Y \to R X$ is a weak equivalence. By 2-out-of-3, the induced map $X \to R X \times_{R Y} Y$ is also a weak equivalence.

Now by Ken Brown's lemma, the pullback functor along $Z\to Y$ preserves weak equivalences between fibrant objects, and in particular preserves this weak equivalence $X \to R X \times_{R Y} Y$. Thus, the induced map $X\times_Y Z \to R X \times_{R Y} Z$ is a weak equivalence. However, $R X \times_{R Y} Z \to R X$ is a weak equivalence by the assumption, so by 2-out-of-3, the map $X\times_Y Z \to X$ is also a weak equivalence, as desired.

### Homotopy (co)limits in proper model categories

###### Proposition

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

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

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

The standard procedure to produce this $X'$ 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 $\emptyset \hookrightarrow K' \stackrel{\simeq}{\to}$ of $K$ and factor $K' \to K \to Y$ as a cofibration followed by a weak equivalence $K' \hookrightarrow Y' \stackrel{\simeq}{\to} Y$ and similarly factor $K' \to K \to L$ as $K' \hookrightarrow L' \stackrel{\simeq}{\to} L$

This yields a weak equivalence of diagrams

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

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

$X' \coloneqq L' \coprod_{K'} Y'$

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

$\array{ K &&&\to&&& Y \\ & \nwarrow^{\mathrlap{\in W}} &&&& \nearrow_{\mathrlap{\simeq}} & \\ && K' &\to& Y' && \\ \downarrow^{\mathrlap{\in cof}} && \downarrow^{\mathrlap{\in cof}} && \downarrow^{\mathrlap{\in cof}} && \downarrow \\ && L' &\to& X' && \\ & {}^{\mathllap{\in W}} \swarrow &&&& \searrow^{\mathrlap{\simeq}} & \\ L'' \coloneqq K \coprod_{K'} L &&&\to&&& L'' \coprod_{K} Y \\ \downarrow^{\mathrlap{\in W}} &&&&&& \downarrow \\ L &&&\to&&& X } \,,$

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

$X' \stackrel{\simeq}{\to} L'' \coprod_K Y \,.$

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

$\array{ K &\to& Y \\ \downarrow^{\mathrlap{\in cof}} && \downarrow \\ L'' &\to& L'' \coprod_{K'} Y \\ \downarrow^{\mathrlap{\in W}} && \downarrow \\ L &\to& X } \,.$

Notice that the top square is a pushout by construction, and the total one by assumption. Therefore by the pasting law, 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

$\array{ K &\stackrel{\in cof}{\to}& Z &\stackrel{\in W}{\to}& Y \\ \downarrow^{\mathrlap{\in \cof}} && \downarrow^{\mathrlap{\in cof}} && \downarrow \\ L'' &\stackrel{\in cof}{\to}& L'' \coprod_{K} Z &\stackrel{\in W}{\to}& L'' \coprod_{K'} Y \\ \downarrow^{\mathrlap{\in W}} && \downarrow^{\mathrlap{\in W}} && \downarrow \\ L & \to& L \coprod_K Z &\stackrel{\in W}{\to}& X } \,.$

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

$L'' \coprod_{K'} Y \stackrel{\simeq}{\to} X$

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 prop. 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 $A \hookrightarrow C' \stackrel{\simeq}{\to} C$ by a cofibration followed by an acyclic fibration. This give a weak equivalence of pushout diagrams

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

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 $hocolim \stackrel{\simeq}{\to} B \coprod_A C$

$\array{ A &\stackrel{\in cof}{\to}& C' &\stackrel{\in W \cap fib}{\to}& C \\ \downarrow^{\mathrlap{\in cof}} && \downarrow^{\mathrlap{\in cof}} && \downarrow^{\mathrlap{\in cof}} \\ B &\to& hocolim &\stackrel{\in W}{\to}& B \coprod_A C } \,.$

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

###### Proposition

A model category is right proper if and only if every fibration is a sharp map.

(Rezk 98)

### Slice categories

For any model category $M$, and any morphism $f \colon A\to B$, the base change of slice categories along $f$

$\Sigma_f \;\colon\; M/A \rightleftarrows M/B \;\colon\; f^*$

is a Quillen adjunction between slice model categories (this Prop.).

If this adjunction is a Quillen equivalence, then $f$ must be a weak equivalence. In general, the converse can be proven only if $A$ and $B$ are fibrant.

###### Proposition

The following are equivalent:

1. $M$ is right proper.

2. If $f$ is any weak equivalence in $M$, then $\Sigma_f \dashv f^*$ is a Quillen equivalence.

This is due to Rezk 02, Prop. 2.5.

###### Remark

The statement of Prop. may be read as saying $M$ is right proper iff all slice model categories have the “correct” Quillen equivalence type.

Since whether or not a Quillen adjunction is a Quillen equivalence depends only on the classes of weak equivalences, not the fibrations and cofibrations, it follows that being right proper is really a property of a homotopical category. In particular, if one model structure is right proper, then so is any other model structure on the same category with the same weak equivalences.

### Local cartesian closure

Since most well-behaved model categories are equivalent to a model category in which all objects are fibrant — namely, the model category of algebraically fibrant objects — they are in particular equivalent to one which is right proper. Thus, right properness by itself is not a property of an $(\infty,1)$-category, only of a particular presentation of it via a model category.

However, if a Cisinski model category is right proper, then the $(\infty,1)$-category which it presents must be locally cartesian closed. Conversely, any locally cartesian closed (∞,1)-category has a presentation by a right proper Cisinski model category; see locally cartesian closed (∞,1)-category for the proof.

## References

The concept originates in

• Aldridge Bousfield, Eric Friedlander, def. 1.1.6 in Homotopy theory of $\Gamma$-spaces, spectra, and bisimplicial sets, Springer Lecture Notes in Math., Vol. 658, Springer, Berlin, 1978, pp. 80-130. (pdf)

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

The general theory can be found in

also in

Proposition can be found in