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# Macquarie University  Department of Mathematics

## Workshop on Categorical Methods in Algebra, Geometry and Mathematical Physics

### Satellite to the StreetFest conference in honour of Ross Street's sixtieth birthday

#### July 18-21 2005, Australian National University, Canberra

Thu, 21 July:  11:00 - 11:40

##### Weak units and homotopy 3-types.
###### Kock, Joachim (Universitat Autònoma de Barcelona)

\thanks{Joint work with A.\ Joyal.}
It is shown that every braided
monoidal category arises as $\mathrm{End}(I)$ 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 simply
connected homotopy $3$-types. The proof has a clear conceptual
content and relies on a geometrical argument with string diagrams
and configurations spaces.

Typeset PDF of this abstract.