Homotopy types of strict 3-groupoids

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 $n\ge 3$). 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.