Categories enriched on two sides

✍ Scribed by Max Kelly; Anna Labella; Vincent Schmitt; Ross Street

• Publisher: Elsevier Science
• Year: 2002
• Tongue: English
• Weight: 330 KB
• Volume: 168
• Category: Article
• ISSN: 0022-4049
• DOI: 10.1016/s0022-4049(01)00048-2

✦ Synopsis

We introduce morphisms V → W of bicategories, more general than the original ones of BÃ enabou. 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 BÃ enabou, 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 deÿned, and we examine the structures associated with them.