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Stream: learning: questions

Topic: Indexed enriched categories

James Deikun (Apr 13 2025 at 00:22):

Mike Shulman in Enriched indexed categories says "We develop a theory of categories which are simultaneously (1) indexed over a base category S with finite products, and (2) enriched over an S-indexed monoidal category V."

Has anyone done the opposite, that is, categories that are simultaneously (1) enriched over a monoidal category $V$ and (2) indexed over a $V$-enriched category $S$? Note the difference in relationship between $S$ and $V$.

James Deikun (Apr 13 2025 at 01:19):

I found this but it's only slides and the fact that $\mathrm{\mathcal V\text -Cat}$ is not treated as some kind of enriched thingy is suspicious to me.

Nathanael Arkor (Apr 13 2025 at 06:26):

Do you have a structure you think ought to be an example?

James Deikun (Apr 13 2025 at 15:09):

I can probably get by with variations on the "comodules fibered over comonoids" schema.

Matteo Capucci (he/him) (Apr 14 2025 at 08:50):

James Deikun said:

Has anyone done the opposite, that is, categories that are simultaneously (1) enriched over a monoidal category $V$ and (2) indexed over a $V$-enriched category $S$? Note the difference in relationship between $S$ and $V$.

Maybe https://arxiv.org/abs/1801.01386 ?

James Deikun (Apr 14 2025 at 11:20):

From the abstract it sounds like enriching, say, a fibration in an opfibration, which is basically the same thing as Shulman.