Subsection 8.9. (0MTL) · ScalingStacks

8.9. Categories of diagrams[0MTL]

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

f\colon N({\operatorname{\mathbf{M}}}^{I})\approx N({\operatorname{\mathbf{M}}})^{\discnerve I}\rightarrow N^{f}({\operatorname{\mathbf{M}}})^{\discnerve I},

(8.10)

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 ${\operatorname{\mathbf{M}}}$ .

[05V5]

Theorem 8.11. The map $f$ of (8.10) is a Reedy weak equivalence when ${\operatorname{\mathbf{M}}}={\operatorname{\mathcal{S}}}^{{\operatorname{\mathbf{J}}}}$ , where ${\operatorname{\mathcal{S}}}$ denotes the category of simplicial sets and ${\operatorname{\mathbf{J}}}$ is a small indexing category.

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

[05V6]

Corollary 8.12. There is a natural weak equivalence $N({\operatorname{\mathbf{M}}}^{I})\xrightarrow{\sim}N^{f}({\operatorname{\mathbf{M}}})^{N(I)}$ of complete Segal spaces if ${\operatorname{\mathbf{M}}}={\operatorname{\mathcal{S}}}^{J}$ and $I$ and $J$ are small categories.

We prove (8.11) below.

[05V7]

Remark 8.13. It seems that the theorem of Dwyer and Kan, and hence the statements of (8.11) and (8.12) should hold for any “reasonable” model category ${\operatorname{\mathbf{M}}}$ , where the class of “reasonable” closed model categories includes at least the “cofibrantly generated” simplicial closed model categories. We hope that future work will provide a generalization of these theorems to arbitrary closed model categories.

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:

[05V8]

Theorem 8.14 (Dwyer-Kan). Let $I$ be a small category. The natural map

\class({\operatorname{\mathcal{S}}}^{I})\approx\lim{}_{([k]\rightarrow I)\in\boldsymbol{\Delta}^{\operatorname{op}}I}\class({\operatorname{\mathcal{S}}}^{[k]})\rightarrow\holim_{([k]\rightarrow I)\in\boldsymbol{\Delta}^{\operatorname{op}}I}\class({\operatorname{\mathcal{S}}}^{[k]})^{f}

is a weak equivalence, where $X^{f}$ denotes the fibrant replacement of a space $X$ , and $\holim$ is the homotopy inverse limit construction of [BK72].

[05V9]

Proof. That this map is a weak equivalence from each component of $\class({\operatorname{\mathcal{S}}}^{I})$ to the corresponding component of the homotopy limit follows from [DK84a, 3.4(iii)]. That the map is surjective on path components is a consequence of Proposition 3.4 and Theorem 3.7 of [DK84b]. ∎

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

[05VA]

Lemma 8.15. Let $I$ be a small category and let $W$ be a Reedy fibrant simplicial space. Then the natural map

\Map_{s{\operatorname{\mathcal{S}}}}(\discnerve I,W)\approx\lim{}_{([k]\rightarrow I)\in\boldsymbol{\Delta}^{\operatorname{op}}I}W_{k}\rightarrow\holim_{([k]\rightarrow I)\in\boldsymbol{\Delta}^{\operatorname{op}}I}W_{k}

is a weak equivalence.

[05VB]

Proof. Let $A$ be an object in $s(s{\operatorname{\mathcal{S}}})$ (i.e., a simplicial object in $s{\operatorname{\mathcal{S}}}$ ) defined by

A(m)=\coprod_{[k_{0}]\rightarrow\dots\rightarrow[k_{m}]\in I}F(k_{0})\in s{\operatorname{\mathcal{S}}}.

There is an augmentation map $A(0)\rightarrow\discnerve I$ , and the induced map $\diag^{\prime}{A}\rightarrow\discnerve I$ is a Reedy weak equivalence in $s{\operatorname{\mathcal{S}}}$ , where $\diag^{\prime}\colon s(s{\operatorname{\mathcal{S}}})\rightarrow s{\operatorname{\mathcal{S}}}$ denotes the prolongation of the diagonal functor, in this case defined by $(\diag^{\prime}A)_{n}\approx\diag\left([m]\rightarrow A(m)_{n}\right)$ . The result follows from isomorphisms

\Map_{s{\operatorname{\mathcal{S}}}}(\diag^{\prime}{A},W)\approx\Tot(\Map_{s{\operatorname{\mathcal{S}}}}(A({-}),W))\approx\holim_{[k]\rightarrow I\in\boldsymbol{\Delta}^{\operatorname{op}}I}W_{k},

and the fact that $\Map_{s{\operatorname{\mathcal{S}}}}(\discnerve I,W)\rightarrow\Map_{s{\operatorname{\mathcal{S}}}}(\diag^{\prime}{A},W)$ is a weak equivalence since $W$ is Reedy fibrant. ∎

[05VC]

Proof of (8.11). Using (8.15) we can reinterpret (8.14) as stating that there is a weak equivalence

\class({\operatorname{\mathcal{S}}}^{I})\xrightarrow{\sim}\Map_{s{\operatorname{\mathcal{S}}}}(\discnerve I,N^{f}({\operatorname{\mathcal{S}}})).

Substituting $[m]\times I$ for $I$ in the above for all $m\geq 0$ leads to a Reedy weak equivalence

N({\operatorname{\mathcal{S}}}^{I})\xrightarrow{\sim}N^{f}({\operatorname{\mathcal{S}}})^{\discnerve I},

which is the special case of (8.10) with ${\operatorname{\mathbf{M}}}={\operatorname{\mathcal{S}}}$ . To obtain the case of ${\operatorname{\mathbf{M}}}={\operatorname{\mathcal{S}}}^{J}$ , note that by what we have just shown the maps in

N({\operatorname{\mathcal{S}}}^{I\times J})\xrightarrow{\sim}N^{f}({\operatorname{\mathcal{S}}})^{\discnerve(I\times J)}\approx N^{f}({\operatorname{\mathcal{S}}})^{\discnerve I\times\discnerve J}\xleftarrow{\sim}N^{f}({\operatorname{\mathcal{S}}}^{J})^{\discnerve I}

must be Reedy weak equivalences. ∎

Charles Rezk

Original source: arXiv:math/9811037v3

Original source · math/9811037v3