Category Theory
Zulip Server
Archive

You're reading the public-facing archive of the Category Theory Zulip server.
To join the server you need an invite. Anybody can get an invite by contacting Matteo Capucci at name dot surname at gmail dot com.
For all things related to this archive refer to the same person.

Stream: theory: category theory

Topic: Do we need a full enrichment for day convolution

Brendan Murphy (Mar 16 2024 at 19:17):

The fundmental example of day convolution in my mind is the tensor product of graded objects, where it looks very much like a convolution. And when I'm thinking about graded objects I'm often working in an additive context, so e.g. graded abelian groups. But $\operatorname{gr}\mathsf{Ab} := \operatorname{Fun}(\operatorname{disc}(\Z), \mathsf{Ab})$ doesn't acquire a day convolution monoidal structure as written, because $\operatorname{disc}(\Z)$ is an ordinary category and what I've written is an ordinary functor category, not an $\mathsf{Ab}$-enriched one. This isn't really a problem because we can take the free $\mathsf{Ab}$-enriched category on $\operatorname{disc}(\Z)$. But is there a way to get a day convolution type monoidal structure without this step, e.g. instead using that $\mathsf{Ab}$ is tensored over $\mathsf{Set}$?

Brendan Murphy (Mar 16 2024 at 19:19):

Maybe the thing to do here is to note $\operatorname{Fun}(\operatorname{disc}(\Z), \mathsf{Ab})$ can be identified with the category of cocontinuous functors $\operatorname{Fun}^L(\operatorname{Psh}(\operatorname{disc}(\Z)), \mathsf{Ab})$? And then observe this is an internal hom of presentable categories and do some magic to transport over the day convolution structure?

Brendan Murphy (Mar 16 2024 at 19:22):

It would be nicer if this were a tensor product of presentable categories, so we could view the monoidal structure as a "base change" from ordinary (monoidal) categories to $\mathsf{Ab}$-enriched (monoidal) categories

Brendan Murphy (Mar 16 2024 at 19:30):

Ah wait, $\operatorname{Fun}(\operatorname{disc}(\Z), \mathsf{Set})^{\mathrm{op}}$ is the free completion of $\operatorname{disc}(\Z)$ so $\operatorname{Fun}^R(\operatorname{Fun}(\operatorname{disc}(\Z), \mathsf{Set})^{\mathrm{op}}, \mathsf{Ab})$ can be identified with $\operatorname{Fun}(\operatorname{disc}(\Z), \mathsf{Ab})$. And it's also the tensor product $\operatorname{Fun}(\operatorname{disc}(\Z), \mathsf{Set}) \otimes \mathsf{Ab}$. Well I'm satisfied for now but I'd be happy to hear anyone else's thoughts about day convolution without an enrichment

Nathanael Arkor (Mar 17 2024 at 00:55):

If $C$ is a promonoidal category and $D$ is a multicategory, then the category of functors $[C, D]$ between their underlying categories acquires a multicategory structure, which is monoidal when $D$ is representable and monoidally cocomplete. In particular, this captures Day's convolution structure, taking $D = \mathrm{Set}$. The reference for this is §2.8 of Pisani's Sequential multicategories.

Brendan Murphy (Mar 17 2024 at 00:56):

Oh cool! Thanks for the reference