ScalingStacks

Because the usual examples of closed model categories are not small categories, their classification diagrams are not bisimplicial sets. We may elude this difficulty by positing, after Grothendieck, the existence of a universe $U$ (a model for set theory) in which ${\operatorname{\mathbf{M}}}$ is defined. Then $N({\operatorname{\mathbf{M}}},{\operatorname{\mathbf{W}}})$ is an honest simplicial space (though not modeled in the universe $U$, but rather in some higher universe $U^{\prime}$).

Alternately, we note that there is no difficulty if the model category ${\operatorname{\mathbf{M}}}$ is a small category, and that such exist in practise. As an example, choose an uncountable cardinal $\gamma$, and let ${\operatorname{\mathcal{S}}}_{\gamma}$ denote a skeleton of the category of all simplicial sets which have fewer than $\gamma$ simplices. Then ${\operatorname{\mathcal{S}}}_{\gamma}$ is a small category, and is in fact a simplicial closed model category. (Of course, the category ${\operatorname{\mathcal{S}}}_{\gamma}$ is not suitable for all purposes; for example, it is not cartesian closed.)

If ${\operatorname{\mathbf{M}}}$ is a simplicial model category, and $X$ and $Y$ objects in ${\operatorname{\mathbf{M}}}$, we write $\map_{{\operatorname{\mathbf{M}}}}(X,Y)$ for the function complex from $X$ to $Y$.

We prove (8.3) below.

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 $X$.

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. (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. (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. (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

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

is naturally weakly equivalent to the union of those components of

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

Let ${\operatorname{\mathbf{M}}}$ be a closed model category, and let $I$ denote a small indexing category; recall that the weak equivalences in the category ${\operatorname{\mathbf{M}}}^{I}$ of functors are the object-wise weak equivalences. Consider

where the isomorphism on the left-hand side is that described in (3.11), and the map on the right-hand side is that induced by the Reedy fibrant replacement of $N({\operatorname{\mathbf{M}}})$. If $f$ can be shown to be a weak equivalence, then this means we can compute the homotopy type of the classification diagram associated to ${I}$-diagrams in ${\operatorname{\mathbf{M}}}$ knowing only the homotopy type of the classification diagram of ${\operatorname{\mathbf{M}}}$ itself. In particular, knowing $N({\operatorname{\mathbf{M}}})$ determines the homotopy category $\ho({\operatorname{\mathbf{M}}}^{I})$ of the category of $I$-diagrams in ${\operatorname{\mathbf{M}}}$ for every small category $I$.

A result of Dwyer and Kan shows that this holds at least for certain cases of ${\operatorname{\mathbf{M}}}$.

Taken together with (3.11) we obtain the following corollary.

We prove (8.11) below.

Let $\boldsymbol{\Delta}^{\operatorname{op}}I$ denote the category of simplices of $I$. This is a category in which the objects are functors $f\colon[m]\rightarrow I$, and the morphisms $(f\colon[m]\rightarrow I)\rightarrow(g\colon[n]\rightarrow I)$ consist of functors $\delta\colon[n]\rightarrow[m]$ making $f\circ\delta=g$. The actual theorem of Dwyer and Kan [DK84a], [DK84b] is the following:

To derive (8.11) from (8.14) we use the following lemma.