Weak units and homotopy 3-types
Abstract
We show that every braided monoidal category arises as for a weak unit 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.
Cite
@article{arxiv.math/0602084,
title = {Weak units and homotopy 3-types},
author = {André Joyal and Joachim Kock},
journal= {arXiv preprint arXiv:math/0602084},
year = {2010}
}
Comments
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