University of British Columbia, Vancouver,
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 T-algebras 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 n-category 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 2-functors 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 Weber's proof is different from mine (which remains unpublished), and probably shorter.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.