Aspects of multiple categories

Abstract

The equivalence between the category of double categories with connections and the category of 2-categories was proved by C.P. Spencer and Y.L. Wong. In this work we try to generalize this result i.e. to prove that there is an equivalence between the category of w-categories with connections and the category of co-categories. This we have not done, though we have quite a lot of information on on the general case. We however managed to get a clear equivalence between triple categories with connection and 3-categories. In particular, we have Theorem: The functors 7, A form an adjoint equivalence 7: 3-~---+ 3-e: A where 3-~ is the category of triple categories with connections and 3-e is the category of 3-categories. In chapter II we explore the equivalence between w-categories and co-categories and get information as much as possible on this equivalence. In fact we define a functor 7: w-ea-t---+ co-eat. where w-ea-t denotes the category of w-categories and co-eat. denotes the category of co-categories. Also we define an operation~ (we call it folding operation) in an w-category G and prove that this operation transforms an element x e G into an element of the associated co-category 7G. The key problem which stands as an obstacle from establishing the equivalence in the general case is to find a good formula for the composition~(~ o1 i> in G for n > 3.

Date of Award	Sept 1989
Original language	English
Awarding Institution	Bangor University