On non-strict notions of $n$-category and $n$-groupoid via multisimplicial sets

Abstract

In this paper we first give a simplicial approach to the definition of a non strict $n$-category that we call an $n$-nerve following the idea that a category could be interpreted as a simplicial set, and we prove that our construction generalises the case of the usual non strict 2-category. Next we give a simplicial definition of a non strict $n$-groupoid. Then we associate to any space $X$ an $n$-groupoid $\Pi _{_{n}}(X)$ which generalises the famous Poincar\'e groupoid $\Pi _{_{1}}(X)$ and embodies the $n$-truncated homotopy type of $X$. We also give a natural construction for the geometric realisation of an $n$-groupoid and we conjecture that the functor geometric realisation is an inverse up to equivalence to the functor $\Pi _{_{n}}(\ )$ from the category of $n$-truncated topological spaces to the category $n$-Gr of $n$-groupoids.