Torsors and Their Classification II

Computing Homotopy Classes in Model Categories

Let again \ca C be some Grothendieck topos and G a group object in \ca C. I eventually want to give a proof of the following theorem.

There is a functorial construction of a simplicial object BG in \ca C and a functorial bijection \tors[G] \isom [*,BG] between the set of G–torsors \tors[G] and the set of simplicial homotopy classes of morphisms *\to BG.

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 \ca C in the homotopy category. Generally, in any model category \ca M (and \ca C embeds into the local model structure on the category {\ca C}^{\op{\Simp}} of simplicial objects in \ca C) we can “calculate” homsets from X to Y in the homotopy category by taking a cofibrant replacement X' of X and a fibrant replacement Y' of Y and taking honest homotopy classes of maps from X' to Y' in \ca M.

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 \ca M be a category with weak equivalences and let X and Y be objects in \ca M. Define a category \coc(X,Y) of cocycles from X to Y whose objects are diagrams 
\begin{tikzcd}[column sep=2ex, row sep=1ex]
& Z \arrow[ld, "\sim"'{sloped,pos=-.1}] \arrow[rd] & \\
X & & Y
with Z\iso X a weak equivalence and whose morphisms are the obvious commuting diagrams.

Remember that any category \ca C has a nerve \nerve(\ca C)\in\sset, its n–simplices are composable chains of n morphisms in \ca C. By a slight abuse of notation I will not distinguish between \coc(X,Y) and \nerve(\coc(X,Y)). A slight remark regarding notation; the geometric realization of \nerve(\ca C) is often called the classifying space of \ca C and is denoted by \class(\ca C). Sometimes I will consider “a model of” \class(\ca C) to be any convenient fibrant replacement of \nerve(\ca C) in \sset. In particular, it makes sense to talk about \pi_0 \class(\ca C) and by another slight abuse of notation I will often just write \pi_0(\ca C).

It turns out that in any model category \class(\coc(X,Y)) is a good model for the mapping space from X to Y as long as X is cofibrant and Y 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 \ca M in which weak equivalences are preserved under products, the homset [X,Y] in the homotopy category can in a natural way be computed as 
[X,Y]\isom \pi_0\coc(X,Y)

The map from right to left is easily described as 
\begin{tikzcd}[column sep=2ex, row sep=1ex]
& Z \arrow[ld, "f"'] \arrow[rd, "g"] & \\
X & & Y
\end{tikzcd} \mapsto [g\circ f^{-1}]
In the other direction, the map is harder to describe. Instead, let’s consider the set \pi(X,Y) of (left) homotopy classes of maps from X to Y. Any map X\to Y trivially defines a cocycle. If two such maps f and g are left homotopic, there is a cylinder object \begin{tikzcd}X \arrow[r, yshift=3pt] \arrow[r, yshift=-3pt] & X\otimes I\end{tikzcd} and a homotopy h\colon X\otimes I\to Y such that 
& X \arrow[equal,dl] \arrow[d] \arrow[dr,"f"] & \\
X & X\otimes I\arrow[l, "\sim"'l] \arrow[r, "h" description] & Y \\
& X \arrow[equal,ul] \arrow[u] \arrow[ur, "g"'] &
This means that these three cocyles lie in the same path component of \coc(X,Y). Hence, we have a well defined map \pi(X,Y)\to \pi_0\coc(X,Y).

If X is cofibrant and Y is fibrant, then the map \pi_0 \coc(X,Y)\to{} [X,Y] is a bijection.

To prove this, consider the commutative diagram 
\pi(X,Y) \arrow[r] \arrow[dr, "\isom"'] & \pi_0\coc(X,Y) \arrow[d] \\
{} & {}[X,Y]
in which the diagonal map is a bijection because X is cofibrant and Y is fibrant. Hence, it suffices to show that our map is surjective, i.e. that any cocycle (f,g)\colon \begin{tikzcd}[column sep=.8em] X & \arrow[l] Z \arrow[r] & Y\end{tikzcd} is in the path component of a cocycle \begin{tikzcd}[column sep=.8em] X \arrow[equal,r] & X \arrow[r] & Y\end{tikzcd}. Factor f into a trivial cofibration j followed by a trivial fibration p to obtain 
\begin{tikzcd}[row sep=1ex]
& Z \arrow[dl,"f"'] \arrow[dr,"g"] \arrow[dd, "j" description] & \\
X & & Y \\
& X' \arrow[ul, "p"] \arrow[dashed, ur, "\varphi"'] &
where the map \varphi exists because Y is assumed to be fibrant. Because X is cofibrant we can find a section \psi of p and obtain a commutative diagram 
\begin{tikzcd}[row sep=1ex]
& X \arrow[dl,equal] \arrow[dr,"\psi\varphi"] \arrow[dd, "\psi" description] & \\
X & & Y \\
& X' \arrow[ul, "p"] \arrow[ur, "\varphi"'] &
which proves the lemma.1

If X'\iso X and Y'\iso Y are weak equivalences, then the induced map 
is a bijection.

To prove this we will describe an inverse function. Let (f,g)\in\coc(X,Y) be a cocycle and think of it as a morphism (f,g): Z\to X\times Y. Factor this map into a trivial cofibration followed by a fibration 
Z \arrow[r, "j"] \arrow[dr, "{(f,g)}"'] & W \arrow[d, "{(p_X, p_Y)}"] \\
& X\times Y
and observe that by 2–out–of–3 p_X is a weak equivalence. Form the pullback 
W' \arrow[r, "\sim"] \arrow[d, "{(p_X^*,p_Y^*)}"'] & W\arrow[d, "{(p_X,p_Y)}"] \\
X'\times Y' \arrow[r, "\sim"] & X\times Y
in which the bottom map is a weak equivalence by our assumption on \ca M and the top map is a weak equivalence by right properness. This then implies that p_X^* is a weak equivalence by 2–out–of–3, i.e. (p_X^*,p_Y^*) defines a cocycle in \coc(X',Y'). 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 X and Y be any two objects in \ca M and let X' be a cofibrant replacement of X and Y' a fibrant replacement of Y. Then we have a commutative diagram 
\pi_0\coc(X,Y)\ar[d, "\isom"'] \ar[r] & {}[X,Y] \ar[d, "\isom"] \\
\pi_0\coc(X',Y') \ar[r, "\isom"] & {}[X',Y']
which proves the theorem.

  1. Note that for this first lemma we haven’t used any assumption on the model category \ca M.