Categories enriched on two sides

Abstract

We introduce morphisms V→ W of bicategories, more general than the original ones of Bénabou. When V= 1 , such a morphism is a category enriched in the bicategory W . Therefore, these morphisms can be regarded as categories enriched in bicategories “on two sides”. There is a composition of such enriched categories, leading to a tricategory Caten of a simple kind whose objects are bicategories. It follows that a morphism from V to W in Caten induces a 2-functor V - Cat → W - Cat, while an adjunction between V and W in Caten induces one between the 2-categories V - Cat and W - Cat. Left adjoints in Caten are necessarily homomorphisms in the sense of Bénabou, while right adjoints are not. Convolution appears as the internal hom for a monoidal structure on Caten. The 2-cells of Caten are functors; modules can also be defined, and we examine the structures associated with them.