
Venue
Category
Theory 2004,
University of British Columbia, Vancouver,
21/7/04
Abstract The standard nerve construction shows how a category can be regarded as a simplicial set with certain properties. So, if T is the theory of categories then the category of Talgebras embeds fully into a presheaf category. It turns out that this is true not only for the theory T of categories, but for all theories T of a particular kind  namely, familially representable monads on presheaf categories. I shall explain what these are and how the embedding works. The importance of this is as follows. Almost all of the proposed definitions of ncategory are of one of two types:
The result here shows that every definition of the former type is equivalent to a definition of the latter type. Slides In this pdf file (760KB). The main theorem (page 8) has now also been proved by Mark Weber, in section 4 of his paper 'Familial 2functors and parametric right adjoints', available from his web page. Weber's proof is different from mine (which remains unpublished), and probably shorter. He uses a factorization system, whereas I did it by a direct and rather explicit method.
