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Let $B$ be a fibrant simplicial set and let $B^{(1)}$ be its 1st Postnikov object. Let $\mathscr{G}$ denote a groupoid such that $\mathrm{Obj}(\mathscr{G})=B_{0}=B^{(1)}_{0}$ and $\mathrm{Aut}_{\mathscr{G}}(b) = \pi_{1}(B,b)$ for $b \in B_{0}$ and $N(\mathscr{G})$ is nerve of this groupoid.

My question is this: is there a weak equivalence map $B^{(1)} \longrightarrow N(\mathscr{G})$ or $N(\mathscr{G}) \longrightarrow B^{(1)}$ ? Surely these two simplicial object have same the fundamental and homotopy groups; however, does the isomorphism between them induced a weak equivalence map between simplicial objects $B^{(1)}$ and $N(\mathscr{G})$ ?

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  • $\begingroup$ I edited in some minor adjustments, but stopped short of describing the groupoid any differently. It looks like you want a coproduct of connected groupoids, one for each connected component of $B$, to specify the groupoid up to equivalence -- just specifying automorphism groups falls short of that. Or do I misunderstand what you mean? $\endgroup$
    – Todd Trimble
    Aug 10, 2018 at 14:10

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