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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

Tue, 19 July:  15:15 - 15:55

##### Developing the Theory of Weak Complicial Sets --- A Roadmap
###### Verity, Dominic (Macquarie University, Sydney)

Weak complicial sets were first described in Ross'\ 1987 Orientals paper, where they appear in concluding remarks regarding the nature of weak higher dimensional categorical structures. More recently, he dusted off this notion and pointed out that it deserved further study, if only as a way of studying the nerves of weak $\omega$-categories.

In this talk we develop the theory of weak complicial sets, showing in particular that the category of such things may be (strictly) enriched over itself. To this structure we apply a generalisation of the homotopy coherent nerve construction in order to show that the universe of weak complicial sets bears a natural weak complicial structure. We also discuss a simple construction which allows us to derive Tamsamani-like globular $\omega$-categories from weak complicial sets.

More tentatively, we foreshadow a coherence theorem, based on a Yoneda argument, by which it is possible to demonstrate that every weak complicial set is equivalent to the nerve of a (strict) category Gray-enriched in weak complicial sets.

Typeset PDF of this abstract.