arXiv subjects
Publications and source records attributed to Zouhair Tamsamani.
This paper constitutes a recent work using the constructions of a previous preprint alg-geom/9512006 to show that the functors geometric realisation and Poincaré $n$-groupoid induce an equivalence between the category of $n$-grouppoids and the category of $n$-truncated topological spaces, when we localise both categories by weak equivalence.
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 $Π_{_{n}}(X)$ which generalises the famous Poincaré groupoid $Π_{_{1}}(X)$ 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 $Π_{_{n}}(\ )$ from the category of $n$-truncated topological spaces to the category $n$-Gr of $n$-groupoids.