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Daniel Teixeira (Jun 23 2026 at 19:45):I think a V-category is the same as a monad X-->X in the (bi)category of categories and V-valued profunctors between them, where X is a set. Does anyone have a reference for this fact?
Nathanael Arkor (Jun 23 2026 at 19:52):This is essentially how enriched categories are defined in Variation through enrichment, the only difference being that they consider monads in the bicategory of matrices. But this coincides with the sub-bicategory of distributors on discrete categories (at least when discrete enriched categories exist).
Matteo Capucci (he/him) (Jun 24 2026 at 08:35):Wouldn't Mat(V) suffice? In that case, this is observed e.g. in Cruttwell-Shulman's A unified framework for generalized multicategories, though surely before that
Nathanael Arkor (Jun 24 2026 at 09:05):Yes, that's why the bicategory of matrices is used in Variation through enrichment, which is from 1983.
Matteo Capucci (he/him) (Jun 24 2026 at 09:46):Nathanael Arkor said:
the only difference being that they consider monads in the bicategory of matrices
apologies, I missed this part!