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The functors $$\overline {W}$$ and $$\text{Diag}\circ \text{Nerve}$$ are simplicially homotopy equivalent. (English) Zbl 1186.55011
Let $$G$$ be a simplicial group. There are two well-known classifying simplicial set constructions: (1) Kan’s classifying simplicial set $$\overline {W}G$$ and (2) dimensionwise application of the nerve functor for groups yields a bisimplicial set $$NG$$, to which one can apply the diagonal functor to obtain a simplicial set $$\operatorname{Diag} NG$$.
It is well-known that $$\overline {W}G$$ is weakly homotopy equivalent to $$\operatorname{Diag} NG$$. In this article, the author proves that $$\overline {W}G$$ is a strong simplicial deformation retract of $$\operatorname{Diag} NG$$. This gives a stronger relationship between $$\overline {W}G$$ and $$\operatorname{Diag}NG$$.
Reviewer: Jie Wu (Singapore)
##### MSC:
 55U10 Simplicial sets and complexes in algebraic topology 18G30 Simplicial sets; simplicial objects in a category (MSC2010)
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