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The functors Wbar and Diag o Nerve are simplicially homotopy equivalent

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Sebastian Thomas

Given a simplicial group \(G\), there are two known classifying
simplicial set constructions, the Kan classifying simplicial set
\(\KanClassifyingSimplicialSet G\) and \(\DiagonalFunctor \Nerve
G\), where \(\Nerve\) denotes the dimensionwise nerve. They are
known to be weakly homotopy equivalent. We will show that
\(\KanClassifyingSimplicialSet G\) is a strong simplicial
deformation retract of \(\DiagonalFunctor \Nerve G\). In particular,
\(\KanClassifyingSimplicialSet G\) and \(\DiagonalFunctor \Nerve G\)
are simplicially homotopy equivalent.

Journal of Homotopy and Related Structures, Vol. 3(2008), No. 1, pp. 359-378