Homotopy types of strict 3-groupoids

TL;DR

This paper demonstrates that strict n-groupoids cannot model the n-type of a 2-sphere for n≥3 under certain realization functors, and proposes a new concept called 'snucategory' to address this limitation.

Contribution

It shows the limitations of strict n-groupoids in modeling certain homotopy types and introduces the idea of snucategories as a potential solution.

Findings

Strict n-groupoids cannot realize the 2-sphere's n-type for n≥3.

A minimal compatibility condition with homotopy groups is considered.

Proposal of 'snucategory' as a new framework to overcome these limitations.

Abstract

We look at strict $n$-groupoids and show that if $\Re$ is any realization functor from the category of strict $n$-groupoids to the category of spaces satisfying a minimal property of compatibility with homotopy groups, then there is no strict $n$-groupoid $G$ such that $\Re (G)$ is the $n$-type of $S^2$ (for $n\geq 3$). At the end we speculate on how one might fix this problem by introducing a notion of ``snucategory'', a strictly associative $n$-category with only weak units.