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# Weak units and homotopy 3-types

By André Joyal and Joachim Kock

*Street Festschrift: Categories in algebra,
geometry and mathematical physics*,

Contemp. Math **431** (2007), 257-276.

ArXiv:math.CT/0602084.

#### Abstract

We show that every braided monoidal category arises as End(*I*) for
a weak unit *I* in an otherwise completely strict monoidal
2-category. This implies a version of Simpson's weak-unit
conjecture in dimension 3, namely that one-object 3-groupoids
that are strict in all respects, except that the object has only
weak identity arrows, can model all connected, simply connected
homotopy 3-types. The proof has a clear intuitive content and relies
on a geometrical argument with string diagrams and configuration
spaces.
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Last updated: 2007-07-02 by
Joachim Kock.