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Let $\Gamma=(\Gamma_1\rightrightarrows \Gamma_0), \Gamma’=(\Gamma’_1\rightrightarrows \Gamma’_0)$ be Lie groupoids and $\Gamma_{\bullet} ,\Gamma’_{\bullet}$ be the simplicial manifolds associated to $\Gamma,\Gamma’$ respectively.

Question : If the simplicial manifolds $\Gamma_{\bullet}$ and $\Gamma’_{\bullet}$ are isomorphic, then, does it imply that $\Gamma=(\Gamma_1\rightrightarrows \Gamma_0)$ and $ \Gamma’=(\Gamma’_1\rightrightarrows \Gamma’_0)$ are Morita equivalent?

Has this been mentioned anywhere?

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If the simplicial manifolds are isomorphic, then the groupoids are also isomorphic, since the nerve functor from groupoid objects in a category to simplicial objects in the same category is fully faithful.

So to answer your question: yes, but in a boring way.

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  • $\begingroup$ I did not knew about this.... nerve functor from “groupoid objects” in a category to “simplicial objects” in the same category is fully faithful... ok.. $\endgroup$ – Praphulla Koushik Jul 7 '19 at 8:52
  • $\begingroup$ Can you suggest some reference where this has been discussed.. $\endgroup$ – Praphulla Koushik Jul 7 '19 at 10:18
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    $\begingroup$ I think it is in a monograph by Duskin, but the usual proof for small categories and simplicial sets should adapt fine. This is originally due to Grothendieck, but there must be more recent treatments; I'll have to dig around. $\endgroup$ – theHigherGeometer Jul 7 '19 at 12:05
  • $\begingroup$ Is it Simplicial methods and the interpretation of “triple” cohomology or Higher-dimensional torsors and the cohomology of topoi: the abelian theory or some other lecture notes. $\endgroup$ – Praphulla Koushik Jul 7 '19 at 20:13
  • $\begingroup$ I think it's Higher-dimensional torsors and the cohomology of topoi: the abelian theory... $\endgroup$ – theHigherGeometer Jul 7 '19 at 22:14

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