Subsection 8.6. (0MTK) · ScalingStacks

8.6. Results about classification spaces[0MTK]

Recall that the classification space $\class{\operatorname{\mathbf{M}}}$ of a model category is defined to be $\nerve\we({\operatorname{\mathbf{M}}})$ . Given a closed model category ${\operatorname{\mathbf{M}}}$ and an object $X\in{\operatorname{\mathbf{M}}}$ , write $\sclass X$ for the component of $\class({\operatorname{\mathbf{M}}})$ containing $X$ .

[05V2]

Proposition 8.7 (Dwyer-Kan [DK84a, 2.3, 2.4]). Given a simplicial closed model category ${\operatorname{\mathbf{M}}}$ , and an object $X\in{\operatorname{\mathbf{M}}}$ which is both fibrant and cofibrant, let $\haut X\subset\map_{{\operatorname{\mathbf{M}}}}(X,X)$ be its simplicial monoid of weak equivalences. Then the classifying complex $\bar{W}\haut X$ is weakly equivalent to $\sclass X$ ; in fact, $\bar{W}\haut X$ and $\sclass X$ can be connected by a finite string of weak equivalences which is natural with respect to simplicial functors $f\colon{\operatorname{\mathbf{M}}}\rightarrow{\operatorname{\mathbf{N}}}$ between closed model categories which preserve weak equivalences and are such that $fX\in{\operatorname{\mathbf{N}}}$ is both fibrant and cofibrant.

[05V3]

Remark 8.8. We can interpret (8.7) as saying that for any two fibrant-and-cofibrant objects $X,Y\in{\operatorname{\mathbf{M}}}$ , the space of paths from $X$ to $Y$ in $\class({\operatorname{\mathbf{M}}})$ is naturally weakly equivalent to the space $\hoequiv_{{\operatorname{\mathbf{M}}}}(X,Y)\subset\map_{{\operatorname{\mathbf{M}}}}(X,Y)$ of homotopy equivalences from $X$ to $Y$ . (The notation $\class({\operatorname{\mathbf{M}}})$ was defined in (1.2).) Compare with (6.4, 4).

Let ${\operatorname{\mathbf{M}}}$ be a simplicial closed model category. Then ${\operatorname{\mathbf{M}}}^{[n]}$ also admits a simplicial closed model category structure, in which a map $f\colon X\rightarrow Y$ in ${\operatorname{\mathbf{M}}}^{[n]}$ is

(1)

a weak equivalence if $fi\colon Xi\rightarrow Yi$ is a weak equivalence in ${\operatorname{\mathbf{M}}}$ for each $0\leq i\leq n$ ,
(2)

a fibration if $fi\colon Xi\rightarrow Yi$ is a fibration in ${\operatorname{\mathbf{M}}}$ for each $0\leq i\leq n$ , and
(3)

a cofibration if the induced maps $Xi\amalg_{X(i-1)}Y(i-1)\rightarrow Yi$ are cofibrations in ${\operatorname{\mathbf{M}}}$ for each $0\leq i\leq n$ , and we let $X(-1)=Y(-1)$ denote the initial object in ${\operatorname{\mathbf{M}}}$ .

Furthermore, a map $\delta\colon[m]\rightarrow[n]$ induces a functor $\delta^{*}\colon{\operatorname{\mathbf{M}}}^{[n]}\rightarrow{\operatorname{\mathbf{M}}}^{[m]}$ which is simplicial and which preserves fibrations, cofibrations, and weak equivalences.

If $Y$ is a fibrant-and-cofibrant object in ${\operatorname{\mathbf{M}}}^{[n]}$ , with restriction $Y^{\prime}\in{\operatorname{\mathbf{M}}}^{[n-1]}$ formed from the first $n$ objects and $(n-1)$ maps in $[n]$ , then the homotopy fiber of the map

\bar{W}\haut_{{\operatorname{\mathbf{M}}}^{[n]}}Y\rightarrow\bar{W}\haut_{{\operatorname{\mathbf{M}}}^{[n-1]}}Y^{\prime}\times\bar{W}\haut_{{\operatorname{\mathbf{M}}}}Y(n)

is weakly equivalent the union of those components of $\map_{{\operatorname{\mathbf{M}}}}(Y(n-1),Y(n))$ containing conjugates of the given map $Y_{n-1}\colon Y(n-1)\rightarrow Y(n)$ ; by conjugate we mean maps of the form $j\circ Y_{n-1}\circ i$ where $i$ and $j$ are self-homotopy equivalences of $Y(n-1)$ and $Y(n)$ respectively. Here $\bar{W}$ denotes the classifying complex as in [May67, p. 87]. Applying this fibration iteratively shows that the homotopy fiber of the map

\bar{W}\haut_{{\operatorname{\mathbf{M}}}^{[n]}}Y\rightarrow\bar{W}\haut_{{\operatorname{\mathbf{M}}}}Y(0)\times\dots\times\bar{W}\haut_{{\operatorname{\mathbf{M}}}}Y(n)

is naturally weakly equivalent to the union of those components of

\map_{{\operatorname{\mathbf{M}}}}(Y(0),Y(1))\times\dots\times\map_{{\operatorname{\mathbf{M}}}}(Y(n-1),Y(n))

containing “conjugates” of the given sequence of maps $Y_{i}\colon Y(i)\rightarrow Y(i+1)$ .

[05V4]

Proof of (8.3). Let $U=N({\operatorname{\mathbf{M}}})$ , so that $U_{n}=\nerve\we({\operatorname{\mathbf{M}}}^{[n]})$ and $U_{n}\rightarrow V_{n}$ is a weak equivalence of spaces. For each $n\geq 0$ there is a map $\pi_{n}\colon U_{n}\rightarrow U_{0}^{n+1}$ which “remembers” only objects. The remarks above together with (8.7) show that for each $(n+1)$ -tuple of objects $(X_{0},\dots,X_{n})$ in ${\operatorname{\mathbf{M}}}$ the homotopy fiber of $\pi_{n}$ over the point corresponding to $(X_{0},\dots,X_{n})$ is in a natural way weakly equivalent to a product

\map_{{\operatorname{\mathbf{M}}}}(X_{n-1}^{\prime},X_{n}^{\prime})\times\dots\times\map_{{\operatorname{\mathbf{M}}}}(X_{0}^{\prime},X_{1}^{\prime}),

where $X_{i}^{\prime}$ is a fibrant-and-cofibrant object of ${\operatorname{\mathbf{M}}}$ which is weakly equivalent to $X_{i}$ .

Note that it is an immediate consequence of the above that $V$ is a Segal space. Since $\pi_{0}U_{0}$ is just the set of weak homotopy types in ${\operatorname{\mathbf{M}}}$ , and since $\ho{\operatorname{\mathbf{M}}}(X,Y)\approx\pi_{0}\map_{{\operatorname{\mathbf{M}}}}(X^{\prime},Y^{\prime})$ where $X^{\prime}$ and $Y^{\prime}$ are fibrant-and-cofibrant replacements of $X$ and $Y$ respectively, we see that $\ho{\operatorname{\mathbf{M}}}\approx\ho V$ .

Let $U_{\hoequiv}\subset U_{1}$ denote the subspace of $U_{1}$ which corresponds to the subspace $V_{\hoequiv}\subset V_{1}$ . By the equivalence of homotopy categories above, we see that $U_{\hoequiv}$ consists of precisely the components of $U_{1}$ whose points go to isomorphisms in $\ho{\operatorname{\mathbf{M}}}$ . Since ${\operatorname{\mathbf{M}}}$ is a closed model category, this means that the $0$ -simplices of $U_{\hoequiv}$ are precisely the objects of ${\operatorname{\mathbf{M}}}^{[1]}$ which are weak equivalences, so $U_{\hoequiv}=\nerve\we((\we{\operatorname{\mathbf{M}}})^{[1]})$ . There is an adjoint functor pair $F\colon{\operatorname{\mathbf{M}}}^{[1]}\leftrightarrows{\operatorname{\mathbf{M}}}\;{:}\,G$ in which the right adjoint takes $G(X)=\id_{X}$ , and the left adjoint takes $F(X\rightarrow Y)=X$ ; this pair restricts to an adjoint pair $\we((\we{\operatorname{\mathbf{M}}})^{[1]})\leftrightarrows\we{\operatorname{\mathbf{M}}}$ and thus induces a weak equivalence $U_{\hoequiv}\approx U_{0}$ of the nerves. Thus $V$ is a complete Segal space. ∎

Charles Rezk

Original source: arXiv:math/9811037v3

Original source · math/9811037v3