Theorem 8.14 (05V8) · ScalingStacks

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]. ∎