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 $\begin{tikzcd}[column sep=2ex, row sep=1ex] & Z \arrow[ld, "\sim"'{sloped,pos=-.1}] \arrow[rd] & \\ X & & Y \end{tikzcd}$ 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 $\begin{tikzcd} & X \arrow[equal,dl] \arrow[d] \arrow[dr,"f"] & \\ X & X\otimes I\arrow[l, "\sim"'l] \arrow[r, "h" description] & Y \\ & X \arrow[equal,ul] \arrow[u] \arrow[ur, "g"'] & \end{tikzcd}$ 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 $\begin{tikzcd} \pi(X,Y) \arrow[r] \arrow[dr, "\isom"'] & \pi_0\coc(X,Y) \arrow[d] \\ {} & {}[X,Y] \end{tikzcd}$ 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 $\begin{tikzcd}[row sep=1ex] & Z \arrow[dl,"f"'] \arrow[dr,"g"] \arrow[dd, "j" description] & \\ X & & Y \\ & X' \arrow[ul, "p"] \arrow[dashed, ur, "\varphi"'] & \end{tikzcd}$ 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 $\begin{tikzcd}[row sep=1ex] & X \arrow[dl,equal] \arrow[dr,"\psi\varphi"] \arrow[dd, "\psi" description] & \\ X & & Y \\ & X' \arrow[ul, "p"] \arrow[ur, "\varphi"'] & \end{tikzcd}$ which proves the lemma.¹

If $X'\iso X$ and $Y'\iso Y$ are weak equivalences, then the induced map $\pi_0\coc(X',Y')\to\pi_0\coc(X,Y)$ 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 $\begin{tikzcd} Z \arrow[r, "j"] \arrow[dr, "{(f,g)}"'] & W \arrow[d, "{(p_X, p_Y)}"] \\ & X\times Y \end{tikzcd}$ and observe that by 2–out–of–3 $p_X$ is a weak equivalence. Form the pullback $\begin{tikzcd} W' \arrow[r, "\sim"] \arrow[d, "{(p_X^*,p_Y^*)}"'] & W\arrow[d, "{(p_X,p_Y)}"] \\ X'\times Y' \arrow[r, "\sim"] & X\times Y \end{tikzcd}$ 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 $\pi_0\coc(X,Y)\to\pi_0\coc(X',Y')$ 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 $\begin{tikzcd} \pi_0\coc(X,Y)\ar[d, "\isom"'] \ar[r] & {}[X,Y] \ar[d, "\isom"] \\ \pi_0\coc(X',Y') \ar[r, "\isom"] & {}[X',Y'] \end{tikzcd}$ which proves the theorem.

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