Categories and their Applications,
Abstract A mature version of the coherence theorem for bicategories should not be limited to 'every weak 2-category is equivalent to a strict one': it should also say something about the functors etc. between 2-categories. With this in mind, I will discuss the various ways to collect together (strict or weak) 2-categories to form a single structure, and what coherence does and does not say about how these structures are related. For instance, if Str2Cat denotes the 3-category of strict 2-categories, strict 2-functors etc., and similarly Wk2Cat (everything weak), then the inclusion Str2Cat → Wk2Cat is not an equivalence. This is a precise expression of the view (long advocated by Bénabou) that the most important aspect of the theory of bicategories is not that they themselves are weak, but that the maps between them are weak.
Although this talk will consist of elementary observations, I will assume knowledge of basic bicategory theory: see for instance the references below.
Slides In this pdf file (700KB).