Categories enriched on two sides

Max Kelly, Anna Labella, Vincent Schmitt, Ross Street*

*Corresponding author for this work

    Research output: Contribution to journalArticlepeer-review

    12 Citations (Scopus)


    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.

    Original languageEnglish
    Pages (from-to)53-98
    Number of pages46
    JournalJournal of Pure and Applied Algebra
    Issue number1
    Publication statusPublished - 8 Mar 2002


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