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Stream: theory: category theory

Topic: internal monoid actions

Morgan Rogers (he/him) (Oct 06 2022 at 12:29):

Monoids in monoidal categories arise frequently when one is learning about monoidal categories. However, one much more rarely sees actions of such monoids discussed. As is typical for me, I'm considering the category of actions of a generic monoid in a monoidal category. I'm struggling to find references where these are discussed. Actions of monoidal categories seem to have received plenty of attention, which makes searching difficult. I came across a paper of H.E. Porst from 1987 but it's paywalled. Has anyone seen something along these lines, even for slightly specialized cases (eg. groups instead of monoids, or cartesian monoidal categories)?

Morgan Rogers (he/him) (Oct 06 2022 at 12:42):

Perhaps there is someone who has studied these using string diagrams?

Dylan Braithwaite (Oct 06 2022 at 12:46):

Have you looked at the references on [[module object]] already?

Dylan Braithwaite (Oct 06 2022 at 12:52):

(deleted)

Morgan Rogers (he/him) (Oct 06 2022 at 12:55):

That's helpful, it looks like the name of what I was looking for, although the more modern references are still a bit removed from what I was after. Maybe I'll check what Grothendieck did :')

Morgan Rogers (he/him) (Oct 06 2022 at 13:19):

To be honest I think calling them "module objects" is misleading since it invokes a somewhat special case. Maybe I'll have some influence on that.

Joe Moeller (Oct 06 2022 at 13:32):

I think this is related to how some people commonly call monoid objects "algebras". I was even reading a paper yesterday that calls them "ring objects". I really thought they meant ring object until later when they said something like "we must consider ring endofunctors, more commonly called monads".

Morgan Rogers (he/him) (Oct 06 2022 at 13:33):

Gross! Words mean things!! :triumph:
Moderator note: this comment resulted in the creation of this stream/topic.

Nathanael Arkor (Oct 06 2022 at 14:19):

I've seen these studied in the more general context of actions in an $\mathbf M$-actegory (for a monoid $M$ in a monoidal category $\mathbf M$), where they are called "action objects" or just "actions". (You recover an action in $\mathbf M$ by taking $\mathbf M$ to be the trivial $\mathbf M$-actegory.) I notice this generalisation is mentioned on the page [[module object]], but it may be worth explicitly mentioning that the literature on actegories is a good place to look.

Nathanael Arkor (Oct 06 2022 at 14:21):

(I also prefer the term "action object" over "module object" for this reason – that the terminology is then more consistent – and I find "module" tends to be more overloaded in practice.)

Morgan Rogers (he/him) (Oct 06 2022 at 14:40):

Oof this pursuit is heading for greater generality than I had anticipated in that case.
It seems like the default (cf Borceux-Janelitze-Kelly) is to pursue the reverse order of constructions to what I'm considering: to fix an object being acted upon and consider the collection of all actions of monoids on that object, rather than fixing the monoid and considering its actions on all objects. I'll keep reading and see if they get anywhere near what I'm considering.

Morgan Rogers (he/him) (Oct 06 2022 at 15:11):

They didn't do anything on the thing I'm working on, which is good and bad news. On the bright side, I realised that using actegories enables me to assume even less structure on the category of objects being acted on, which (since my angle is to examine how adding structure to the base category adds structure to the actions) might be a small bonus.

Morgan Rogers (he/him) (Oct 06 2022 at 15:20):

@Matteo Capucci (he/him) and @Bruno Gavranovic since you wrote the recent survey on actegories, have you thought about/seen monoid actions in them? Is this a special case of something in the actegory canon that I'm just not quite seeing? It makes a lot of sense that this would be the "right" setting for them, as an instance of the microcosm principle, but what is out there? (I have found your statement of this observation in Definition 3.1.9)

Nathanael Arkor (Oct 06 2022 at 15:27):

It's also probably worth noting that many papers (especially older ones) don't use the term "actegory" and instead use terms like "action on a monoidal category", which makes it more difficult to search for the concept.

Morgan Rogers (he/him) (Oct 07 2022 at 12:25):

It occurs to me that I could have revived #theory: monoids with this topic :joy:

Matteo Capucci (he/him) (Oct 07 2022 at 17:56):

Morgan Rogers (he/him) said:

Matteo Capucci (he/him) and Bruno Gavranovic since you wrote the recent survey on actegories, have you thought about/seen monoid actions in them? Is this a special case of something in the actegory canon that I'm just not quite seeing? It makes a lot of sense that this would be the "right" setting for them, as an instance of the microcosm principle, but what is out there? (I have found your statement of this observation in Definition 3.1.9)

Trivially, everything we say about actegories applies for 'discrete actegories', i.e. monoid actions. We didn't have the heart of writing the paper parametrized by a given cartesian 2-category $\cal K$, so I can't go beyond this and say you can apply all the results to monoid objects in any category.
As you note, we comment on the microcosm phenomenon whereby the most general place where to define 'actions of a monoid' (which I call modules) is an actegory: monoids come from the acting monoidal category $\cal M$ and module objects from the acted upon category $\cal C$. Various dualization of this observation still holds, and we show that lax linear morphisms of actegories preserve modules in the same way lax monoidal functors preserve monoids.

Morgan Rogers (he/him) (Oct 07 2022 at 19:14):

I'll try to find that result!
In any case, today I decided that acts in actegories were the right setting for what I wanted to do and I started writing the paper :innocent:

Matteo Capucci (he/him) (Oct 08 2022 at 18:03):

:D amazing!

Matteo Capucci (he/him) (Oct 08 2022 at 18:05):

(Also, I really like the terminology acts. It retroactively makes this post/talk of mine about optics a pun. Unfortunately, mixed optics only need two acts!)

Tom Hirschowitz (Oct 10 2022 at 08:22):

To get back to the original question: my coauthors and I use modules over monoids, though not very deeply, e.g., in
A categorical framework for congruence of applicative bisimilarity in higher-order languages and Modules over monads and operational semantics. The intuition in our use cases is that a monoid $M$ is something equipped with some form of substitution, and a module is an object equipped with substitution by elements of $M$, i.e., a potentially external form of substitution.

Morgan Rogers (he/him) (Oct 12 2022 at 09:15):

I thought it was a problem with my phone but it seems that the first of those two links is actually broken @Tom Hirschowitz

Tom Hirschowitz (Oct 12 2022 at 20:03):

Woops, sorry, fixed, thanks @Morgan Rogers (he/him).