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Strict n-categories model homotopy types

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Andrea Gagna
Aix-Marseille Université
Mit, 15/02/2017 - 13:30 - 14:00
MPIM Lecture Hall

Let $\mathcal{C}$ be a cocomplete category and $c \colon \mathbf{\Delta} \to \mathcal{C}$ a cosimplicial object. We denote by $N$ the induced nerve functor. We call equivalences of $\mathcal{C}$ the morphisms which are sent to simplicial weak equivalences by $N$. We give a sufficient condition, namely that the unity $1_{\widehat{\mathbf{\Delta}}} \to Nc$ is a weak equivalence for the usual nerve of posets, such that the functor $N \colon C \to \widehat{\mathbf{\Delta}}$ induces an equivalence at the level of the homotopy categories. As this condition has been verified for the (truncated) oriental functor $\mathcal{O}\colon \mathbf{\Delta} \to m\text{-}\mathcal{C}\textit{at} $ by D. Ara and G. Maltsiniotis for $1\leqslant m \leqslant \infty$, we have that the Street's nerve $N_m \colon m\text{-}\mathcal{C}\textit{at} \to \widehat{\mathbf{\Delta}}$ is an equivalence in homotopy.

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