The ziqqurath of exact sequences of n-groupoids

TL;DR

This paper introduces a new notion of exactness for sequences of pointed n-groupoids, generalizing classical results and constructing a hierarchical 'ziqqurath' structure of higher groupoids and sets.

Contribution

It develops a framework for exact sequences of pointed n-groupoids and generalizes a known 6-term exact sequence from groupoid fibrations to higher dimensions.

Findings

Established a notion of exactness for n-groupoid sequences

Generalized the 6-term exact sequence to higher groupoids

Constructed the hierarchical 'ziqqurath' structure of n-groupoids

Abstract

Higher Dimensional Categories are showing relevant implications in several fields of mathematical research. Nevertheless basic algebraic tools, in order to further develop the theory, are far from being established. In this thesis we introduce a notion of exactness for exact sequences of pointed n-groupoids. Furthermore we test it generalizing a well known result for (fibrations of) groupoids [R.Brown, 1970]. Namely, given a fibration F of (pointed) groupoids and its strict kernel it is possible to obtain a 6-term exact sequence of groups (of loops) and pointed sets (iso classes of objects). The ziqqurath, aka step-pyramid, comes out from iterating this construction, and it consists in several sequences of n-groupoids, (n-1)-groupoids and so on up to pointed sets (0-groupoids), of increasing length.