2-operads and bicategories

Michael Batanin – 4 September 1996

I consider in this talk the two dimensional case of the general definition of a weak omega-category (this definition was given in my talks on 14 August 1996 at Macquarie University ) My aim is to demonstrate that the weak 2-categories in the sense of this general definition are bicategories and conversely. For this I investigate thoroughly the structure of contractible 2-operads and prove the following lemma Lemma. The category of pointed contractible 2-operads is isomorphic to the category of pointed nonsymmetric chaotic Cat-operads (where chaotic means that an operad consists of chaotic categories) and hence, to the category of pointed nonsymmetric Set-operads. This lemma allows to prove the following proposition. Proposition. Every algebra over a contractible pointed 2-operad may be endowed with a canonical sructure of a bicategory. Conversely, every bicategory has a canonical structure over the universal pointed contractible 2-operad. The proof uses the coherence theorem for bicategories.