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Weak identity arrows in higher categories

An important aspect of higher category theory is weakening, since strict higher categories are not sufficient for most purposes. For example, strict n-groupoids cannot model all homotopy n-types. Most approaches to higher categories emphasise weakening of the composition laws, i.e. weak associativity. On this background it was very surprising when Simpson conjectured in 1999 that n-groupoids with strict composition but only weak identity arrows should model all homotopy n-types. I have spend a considerable time on this conjecture, setting up a theory for dealing with weak identity arrows in a systematic manner, and succeeded in proving, together with André Joyal, a version of the conjecture in dimension 3, which is the first nontrivial case.


Last updated: 2013-02-19 by Joachim Kock.