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Stream: theory: category theory

Topic: Hinich enriched category theory

Adrian Clough (Oct 02 2025 at 11:46):

In Enriched Yoneda Lemma (https://arxiv.org/abs/1511.00857v2) Hinich presents a very slick version of the enriched Yoneda embedding. One aspect of the paper which I find intriguing is that this version of the Yoneda embeddings is expressed using functors from categories enriched in $\mathcal{M}$ to categories tensored over $\mathcal{M}$, where $\mathcal{M}$ is a (not necessarily symmetric nor closed) monoidal category. I feel like in recent years there has been quite some development of new categorical structures (e.g., virtual equipments etc.) being able to accommodate objects and morphisms “of different types” in the same setting. Does there exist some such structure which naturally captures the interplay between

- $\mathcal{M}$-enriched categories and functors 

- $\mathcal{M}$-tensored categories and functors 

- functors from $\mathcal{M}$-enriched categories to $\mathcal{M}$-tensored categories

as well as weighted/enriched colimits?

Nathanael Arkor (Oct 02 2025 at 12:13):

The general context for this result is the setting of a [[locally graded category]], which was introduced by Richard Wood as a setting in which to compare enriched categories, actegories/tensored categories, and powered categories. Wood's thesis is a good introduction, as is Rory Lucyshyn-Wright's V-graded categories and V-W-bigraded categories: Functor categories and bifunctors over non-symmetric bases.

Nathanael Arkor (Oct 02 2025 at 12:14):

That is, rather than consider some structure with multiple kinds of objects and morphisms, it is most natural to consider all of these structures as special cases of a more general structure: the morphisms between the structures are then exactly the appropriate notion of functors between possibly different kinds of structures.

Adrian Clough (Oct 02 2025 at 13:06):

Cool, thanks, I'll take a look!