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Stream: deprecated: id my structure

Topic: Day Convolution for Bicategories

Max New (Jan 11 2023 at 16:03):

I haven't checked in detail but it looks plausible to me that the Day convolution product on $Set^{C^o}$ for a monoidal category $C$ generalizes to bicategories $C$ where we define a bicategory $\hat C$ with the same objects as $C$ but whose morphisms are presheaves on the hom category $\hat C(a,b) = Set^{C(a,b)^o}$ and the composition in this bicategory is a kind of Day convolution composition. This seems like it would be well-known to bicategory theorists, does anyone know a reference?

Max New (Jan 11 2023 at 16:07):

The main reason I haven't worked it out in this generality is that my situation is much less general: $C$ in my case is just a free category.

Nathanael Arkor (Jan 11 2023 at 16:39):

Yes, this is described in §5 of Kelly–Labella–Schmitt–Street's Categories enriched on two sides, there called local cocompletion of a bicategory.

vikraman (Jan 11 2023 at 17:10):

To add a bit, you can just get it from the promonoidal convolution formula, it's done by Day himself in this paper: https://link.springer.com/content/pdf/10.1007/BFb0063099.pdf?pdf=inline%20link

Nathanael Arkor (Jan 11 2023 at 17:12):

Thanks, that's a better reference.

Christian Williams (Jan 11 2023 at 17:23):

wow, this is a cool idea.

Christian Williams (Jan 11 2023 at 17:23):

how are you using it?

Christian Williams (Jan 11 2023 at 17:42):

Local cocompleteness is what you need to form Mod(K), monads and modules in a bicategory. Normally if K doesn't have all colimits, Mod(K) doesn't have composites, so it is a virtual equipment. But with this construction, we can just freely cocomplete the hom-categories first, and Mod(Day(K)) is always an equipment! This is great.

Nathanael Arkor (Jan 11 2023 at 17:43):

Perhaps it's also worth noting that a more abstract perspective on this construction is as a change-of-enrichment-base for bicategories, from enrichment in the 2-category of small categories, to enrichment in the 2-category of small-cocomplete categories, given by the presheaf construction (this is essentially in §15.8 of Garner–Shulman).

Christian Williams (Jan 11 2023 at 17:51):

Nathanael Arkor said:

(this is essentially in §15.8 of Garner–Shulman).

Well they mention the adjunction and the monad for cocompletion, but I don't see them using it as change-of-base.

Nathanael Arkor (Jan 11 2023 at 17:57):

That was
Christian Williams said:

Nathanael Arkor said:

(this is essentially in §15.8 of Garner–Shulman).

Well they mention the adjunction and the monad for cocompletion, but I don't see them using it as change-of-base.

That was why I included the "essentially" :grinning_face_with_smiling_eyes: They do mention that the biadjunction is monoidal (hence suitable for change-of-base), but don't spell out what change-of-base looks like in that context.