Computing Homotopy Classes in Model Categories
Let again be some Grothendieck topos and a group object in . I eventually want to give a proof of the following theorem.
There is a functorial construction of a simplicial object in and a functorial bijection between the set of –torsors and the set of simplicial homotopy classes of morphisms .
To show this we will follow an approach which was presented in the paper/preprint “Cocyle Categories” by Rick Jardine. The point is that proving our theorem comes down to computing homsets between objects of in the homotopy category. Generally, in any model category (and embeds into the local model structure on the category of simplicial objects in ) we can “calculate” homsets from to in the homotopy category by taking a cofibrant replacement of and a fibrant replacement of and taking honest homotopy classes of maps from to in .
However, computing homsets like that tends to be quite difficult because, typically, cofibrant and fibrant replacement are hard to write down explicitly (at least in a form that one can work with). Jardine came up with the following work–around.
Let be a category with weak equivalences and let and be objects in . Define a category of cocycles from to whose objects are diagrams with a weak equivalence and whose morphisms are the obvious commuting diagrams.
Remember that any category has a nerve , its –simplices are composable chains of morphisms in . By a slight abuse of notation I will not distinguish between and . A slight remark regarding notation; the geometric realization of is often called the classifying space of and is denoted by . Sometimes I will consider “a model of” to be any convenient fibrant replacement of in . In particular, it makes sense to talk about and by another slight abuse of notation I will often just write .
It turns out that in any model category is a good model for the mapping space from to as long as is cofibrant and is fibrant. This is proven for example in a preprint of Daniel Dugger’s, “Classification Spaces of Maps in Model Categories”. I will—following Jardine—give a proof of something weaker:
In a right proper model category in which weak equivalences are preserved under products, the homset in the homotopy category can in a natural way be computed as
The map from right to left is easily described as In the other direction, the map is harder to describe. Instead, let’s consider the set of (left) homotopy classes of maps from to . Any map trivially defines a cocycle. If two such maps and are left homotopic, there is a cylinder object and a homotopy such that This means that these three cocyles lie in the same path component of . Hence, we have a well defined map .
If is cofibrant and is fibrant, then the map is a bijection.
To prove this, consider the commutative diagram in which the diagonal map is a bijection because is cofibrant and is fibrant. Hence, it suffices to show that our map is surjective, i.e. that any cocycle is in the path component of a cocycle . Factor into a trivial cofibration followed by a trivial fibration to obtain where the map exists because is assumed to be fibrant. Because is cofibrant we can find a section of and obtain a commutative diagram which proves the lemma.1
If and are weak equivalences, then the induced map is a bijection.
To prove this we will describe an inverse function. Let be a cocycle and think of it as a morphism . Factor this map into a trivial cofibration followed by a fibration and observe that by 2–out–of–3 is a weak equivalence. Form the pullback in which the bottom map is a weak equivalence by our assumption on and the top map is a weak equivalence by right properness. This then implies that is a weak equivalence by 2–out–of–3, i.e. defines a cocycle in . This construction yields a well–defined map which is inverse to the map in the lemma.
Using these two lemmas the theorem about computing homsets is easy to prove. Let and be any two objects in and let be a cofibrant replacement of and a fibrant replacement of . Then we have a commutative diagram which proves the theorem.
Note that for this first lemma we haven’t used any assumption on the model category .↩