Nonstrict notions of \(n\)-category and \(n\)-groupoid via multisimplical sets
Following the idea that a category could be interpreted as a simplicial set, say, its nerve, the author gives a simplicial approach to the definition of a nonstrict \(n\)-category as an \(n\)-nerve. The author constructs a correspondence which to any topological space associates an \(n\)-groupoid \(\Pi_n(X)\) which generalises the Poincaré groupoid \(\Pi_1(X)\) and embodies the \(n\)-truncated homotopy type of \(X\). It is also shown in this paper that a suitable geometric realisation together with the given Poincaré groupoid, induces an equivalence between the category of \(n\)-groupoids and the category of \(n\)-truncated topological spaces. The paper also provides a proof of equivalence of the usual nonstrict bicategory with the simplicial construction for \(n=2\).
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