Weak units and homotopy 3-types

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.

Dedicated to Ross Street on his 60th birthday; to appear in the StreetFest proceedings. 20 pages, LaTeX; uses Paul Taylor's diagrams and Peter Kabal's texdraw; does not compile with pdflatex. Version v2: reformatted to fit also on letter paper. Version v3: expository improvements, final version