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The notion of category enriched in a bicategory is the many-object-generalization of the notion of an enriched category enriched in a monoidal category via regarding a monoidal category as a bicategory with a single object.
Originally Bénabou called these polyads.
Let be a bicategory, and write for horizontal (1-cell) composition (written in Leibniz order). A category enriched in the bicategory consists of a set together with
such that the associativity and unitality diagrams, as written above, commute. Viewing a monoidal category as a 1-object bicategory , the notion of enrichment in coincides with the notion of enrichment in the bicategory .
Equivalently this is simply a lax functor from the codiscrete category on into . In particular if is the singleton set then this is the same as a monad.
If , are sets which come equipped with enrichments in , then a -functor consists of a function such that , together with a function , satisfying the constraint , and satisfying equations expressing coherence with the composition and unit data , of and . (Diagram to be inserted, perhaps.)
Discussion of Kleisli objects (collages) for monads generalized to categories enriched in bicategories is in section 15.9 of
Last revised on February 12, 2019 at 04:03:57. See the history of this page for a list of all contributions to it.