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Simplicial nerve of a topological group

This is an answer to the edited question. First, observe that the composition of functors $\def\N{{\rm N}}\def\Sing{{\rm Sing}}\N∘\Sing$ in the main post computes the homotopy colimit of the ...
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Simplicial nerve of a topological group

Konrad Waldorf answers this question in his comment, but I didn't want the question to linger in the "unanswered" queue, so here's a CW answer consisting of a page of Segal's 1968 paper that ...
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Is there an analog of Kan's $Ex^\infty$ functor for quasicategories?

$\def\Cnec{{\frak C}^{\rm nec}} \def\Exi{{\rm Ex}^∞} \def\N{{\rm N}} \def\sCat{{\sf sCat}} \def\sSet{{\sf sSet}_{\sf Joyal}}$ As indicated in the answer A combinatorial approximation functor sSet->...
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What is the name for the construction of this poset related to coherence of degeneracies of the simplex category?

Here’s one way to see it, if I’m not misunderstanding your definition. For a small category $\newcommand{\C}{\mathbf{C}}\C$, take its categorical nerve $\newcommand{\N}{\mathbf{N}}\N\C$ to be the ...
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