English

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

alg-geom 2015-06-30 v2 Algebraic Geometry

Abstract

In this paper we first give a simplicial approach to the definition of a non strict nn-category that we call an nn-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 nn-groupoid. Then we associate to any space XX an nn-groupoid Πn(X)\Pi _{_{n}}(X) which generalises the famous Poincar\'e groupoid Π1(X)\Pi _{_{1}}(X) and embodies the nn-truncated homotopy type of XX. We also give a natural construction for the geometric realisation of an nn-groupoid and we conjecture that the functor geometric realisation is an inverse up to equivalence to the functor Πn( )\Pi _{_{n}}(\ ) from the category of nn-truncated topological spaces to the category nn-Gr of nn-groupoids.

Keywords

Cite

@article{arxiv.alg-geom/9512006,
  title  = {On non-strict notions of $n$-category and $n$-groupoid via multisimplicial sets},
  author = {Zouhair Tamsamani},
  journal= {arXiv preprint arXiv:alg-geom/9512006},
  year   = {2015}
}

Comments

Plain TeX, In French. e-mail: [email protected] (with ``for Tamsamani'' in subject)