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Stream: learning: questions

Topic: internal actegory

Joe Moeller (Sep 28 2021 at 15:24):

The definition of an actegory internal to a monoidal bicategory should be pretty straightforward to someone who knows what all those words mean. But has it already been written down somewhere? I'd like to give due credit as well as save myself some work.

John Baez (Sep 28 2021 at 16:13):

In other words, who has talked about something like this: you have a monoidal bicategory with a pseudomonoid $M$ in it, and you want to define and study an "action"

$\alpha: M \otimes X \to X$

of $M$ on some other object $X$.

In the monoidal closed case one should be able to define this in a curried way as a map of pseudomonoids

$\tilde{\alpha} : M \to [X,X],$

at least after one has checked that $[X,X]$ is a pseudomonoid. But probably Joe doesn't want to go down this road....

Joe Moeller (Sep 28 2021 at 16:16):

Well, I do want to go down this road, but it's probably not too worth the effort of defining these things in general when I'm currently just working with one example. But if it does exist in this generality already, that would be fine with me.

John Baez (Sep 28 2021 at 16:30):

I meant going down the road of assuming the monoidal bicategory is monoidal closed!

Joe Moeller (Sep 28 2021 at 16:31):

I know! I think this would be a useful thing to know.

John Baez (Sep 28 2021 at 16:31):

If you do that, you just need to check that $[X,X]$ is always a pseudomonoid and the rest of the definition of action is ready for you. But still it'd be good to find papers where people have already studied these things.

Mike Shulman (Sep 28 2021 at 17:46):

In Enriched categories as a free cocompletion, Richard Garner and I sketched the definition of $V$-enriched category and $V$-enriched presheaf over such, for any monoidal bicategory $V$. Specialized to the one-object case, this gives the notion of pseudomonoid and of module over such. But probably someone has written down a lower-brow version too.

Matteo Capucci (he/him) (Sep 28 2021 at 18:20):

This is my go-to reference for pseudoactions in monoidal bicategories: https://arxiv.org/abs/1801.01386

Matteo Capucci (he/him) (Sep 28 2021 at 18:20):

But this is a secondary source, since iirc Christina cites more extensive references

Joe Moeller (Sep 28 2021 at 18:23):

Seems she cites Day and Street's Monoidal bicategories and Hopf algebroids. I've read the first part of this a bunch, but I guess I forgot they talk about modules since I haven't needed to think about that until now.

Joe Moeller (Sep 28 2021 at 18:24):

Sorry, that's not what she does. She cites DS for pseudomonoid, and then Marmolejo and Lack for the modules.

John Baez (Sep 28 2021 at 21:08):

Cool - Marmolejo and Lack, eh?

There's going to be a category theory day in honor of Marmolejo, maybe you noticed that on the category theory mailing list.