related by the Dold-Kan correspondence
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.
A model category is called
More in detail this means the following. A model category is right proper if for every weak equivalence in and every fibration the pullback in
is a weak equivalence.
by cor. 1, every model category in which all objects are cofibrant is left proper;
this includes notably
the standard model structure on simplicial sets
and many model structures derived from these, such as
So in particular also the local injective model structures on simplicial presheaves over a site are left proper.
A class of model structures which tends to be not left proper are model structures on categories of not-necessarily commutative algebras.
by cor. 1 every model category in which each object is fibrant is right proper.
This includes for instance the standard model structure on topological spaces.
The Quillen model structure on simplicial sets is right proper, though not all objects are fibrant. An explanation of why can be found on its page.
Model categories which are both left and right proper include
The standard model structures on chain complexes.
While some model categories fail to be proper, often there is a Quillen equivalent one that does enjoy properness.
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.
Let be a simplicial (possibly multi-colored) theory, and let 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 such that
This is the content of (Rezk 00)
The following says that left/right properness holds locally in every model category, namely between cofibrant/fibrant objects.
Given a model category,
This gives a large class of examples of left/right proper model categories:
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.
We demonstrate the first statement, the second is its direct formal dual.
So consider a pushout diagram
in a left proper model category, where the morphism is a cofibration, as indicated. We need to exhibit a weak equivalence from an object that is manifestly a homotopy pushout of .
The standard procedure to produce this 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 of and factor as a cofibration followed by a weak equivalence and similarly factor as
This yields a weak equivalence of diagrams
where now the diagram on the right is cofibrant as a diagram, so that its ordinary pushout
is a homotopy colimit of the original diagram. To obtain the weak equivalence from there to , first form the further pushouts
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 is by the 2-out-of-3 property.
This establishes in particular a weak equivalence
It remains to get a weak equivalence further to . For that, take the two outer squares from the above
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 as a cofibration followed by a weak equivalence and push that factorization through the double diagram, to obtain
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
If we had allowed ourselved to assume in addition that itself is already cofibrant, then the above statement has a much simpler proof, which we list just for fun, too.
Let be a cofibration with cofibrant and let be any other morphism. Factor this morphism as by a cofibration followed by an acyclic fibration. This give a weak equivalence of pushout diagrams
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
The proof for the second statement is the precise formal dual.
For any model category , and any morphism , the adjunction
The following are equivalent:
In other words, is right proper iff all slice 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.
See this blog comment.
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 -category, only of a particular presentation of it via a model category.
However, if a Cisinski model category is right proper, then the -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.
The usefulness of right properness for constructions of homotopy categories is discussed in
The general theory can be found in Chapter 13 of