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Stream: learning: questions

Topic: reference for multi-actegories

Nathanael Arkor (Aug 23 2022 at 17:18):

I'm looking for a reference for the generalisation of [[actegories]] from [[monoidal categories]] to [[multicategories]]. Such a structure should comprise a collection of objects, and a collection of multimaps $m_1, \ldots, m_n, a \to a'$ satisfying evident axioms. Does anyone know if these have been defined in the literature before?

Max New (Aug 23 2022 at 20:23):

Like CBPV basically? Following Mike Shulman's terminology (https://arxiv.org/abs/2106.15042) these would be linearly unary, linearly co-unary LNL polycategories, and they can be defined as a slice of the category of LNL polycategories.

Mike Shulman (Aug 23 2022 at 20:58):

Yes, the previous literature on these things seems to have neglected the multicategorical case in favor of assuming the acting object, at least, is a monoidal category. My understanding is that CBPV also usually includes morphisms of the form $m_1,\dots,m_n \to a'$, which corresponds to linearly subunary and co-unary LNL polycategories.

Max New (Aug 23 2022 at 21:20):

Yes, I just call a "linearly subunary and co-unary LNL polycategories" a "CBPV multicategory"

Nathanael Arkor (Aug 23 2022 at 21:21):

Here you're taking $m_1, \ldots, m_n$ to be nonlinear objects and $a, a'$ to be linear objects? Wouldn't this require that the multicategory be cartesian? In any case, I don't think LNL-polycategories are quite what I'm looking for. I can believe it may be possible to encode the structure I'm interested in in that formalism, but I'm more interested in finding in the literature the perspective of actions of multicategories. Furthermore, using LNL-polycategories, I expect that I would be able to capture the concept of "multicategory equipped with a multi-actegory", whereas really I would prefer a notion of multi-actegory for a given (i.e. fixed) multicategory. While this may also be possible by slicing appropriately, I think defining the structure directly is clearer conceptually.

Mike Shulman (Aug 23 2022 at 21:21):

Max New said:

Yes, I just call a "linearly subunary and co-unary LNL polycategories" a "CBPV multicategory"

Of intermediate verbosity is "linearly subunary LNL multicategory".

Max New (Aug 23 2022 at 21:23):

If the NL part is just a multicategory, not cartesian, then I don't know a name for it, though if you included the $m_1,\ldots \to a'$ morphisms I would probably just call it a "linear CBPV multicategory"

Max New (Aug 23 2022 at 21:24):

but no I don't know of any extant names for what you're defining

Mike Shulman (Aug 23 2022 at 21:25):

Nathanael Arkor said:

Here you're taking $m_1, \ldots, m_n$ to be nonlinear objects and $a, a'$ to be linear objects? Wouldn't this require that the multicategory be cartesian?

Yes. But you can deal with the noncartesian case by just using linear objects, which means you can just work with ordinary multicategories. Specifically, there is a symmetric multicategory ACT, with two objects M and A, and such that there is a morphism $\Gamma \to X$ if either (1) $X=M$ and $\Gamma$ consists entirely of Ms, or (2) $X=A$ and $\Gamma$ has exactly one A in it. Then a symmetric multicategory $\mathcal{P}$ with a functor to ACT is precisely a symmetric multicategory (the preimage of M) that "acts" on the objects in the preimage of A. You can do a non-symmetric version too.

Max New (Aug 23 2022 at 21:28):

can you do affine/relevant this way?

Reid Barton (Aug 23 2022 at 21:31):

How about just $M$-actegory, where $M$ is the acting multicategory?

Nathanael Arkor (Aug 23 2022 at 21:40):

Reid Barton said:

How about just $M$-actegory, where $M$ is the acting multicategory?

Do you know any references for this term?

Reid Barton (Aug 23 2022 at 21:41):

No

Mike Shulman (Aug 23 2022 at 22:31):

I don't really like calling this any kind of "actegory", since there isn't actually any "acting" going on. (Actually I don't like the "word" "actegory" in the first place, but that's neither here nor there.)

Reid Barton (Aug 23 2022 at 22:32):

Maybe multiactegory is better? (Though I am not a fan of "actegory" either)

Mike Shulman (Aug 23 2022 at 22:32):

$M$-module?

Reid Barton (Aug 23 2022 at 22:32):

I definitely think this notion deserves a more transparent name than "linearly subunary and co-unary LNL polycategories" or "CBPV multicategory"

Mike Shulman (Aug 23 2022 at 22:34):

Max New said:

can you do affine/relevant this way?

I haven't been able to think of a way to encode affine or relevant structures in terms of cartesian or symmetric ones. You could formulate analogues of LNL polycategories in which the nonlinear objects are affine or relevant instead of cartesian. You could even formulate one "omnibus" version in which there are four classes of objects, cartesian, affine, relevant, and linear. At one point I spent a little while thinking about whether an "entries-only" like formulation could make that tractable, where say each object comes along with the information about the structural rules that it admits.

Nathanael Arkor (Aug 23 2022 at 22:40):

Mike Shulman said:

I don't really like calling this any kind of "actegory", since there isn't actually any "acting" going on. (Actually I don't like the "word" "actegory" in the first place, but that's neither here nor there.)

Calling it an actegory is consistent with the existing terminology for monoidal categories: it's the appropriate generalisation for multicategories. It may not be an action in a formal sense, but I think the terminology "actegory" is merely reminiscent of the structure, which is "action-like".

Nathanael Arkor (Aug 23 2022 at 22:41):

Though I think "multiactegory" is better, since this allows one to disambiguate between the two kinds of "actegory" for representable multicategories.

Nathanael Arkor (Aug 23 2022 at 22:41):

I'm not saying I like the terminology "actegory"; I'm only trying to find the most consistent terminology.

Mike Shulman (Aug 23 2022 at 22:48):

I dunno, if $M$ is a multicategory, I would kind of expect an "$M$-actegory" to be something that reduces to an ordinary actegory when $M$ is a monoidal category. But this doesn't, not until you impose an extra representability condition. What about something like "lax actegory" or "pro-actegory"?

Nathanael Arkor (Aug 23 2022 at 22:49):

That's exactly the reason I was advocating multiactegory rather than just actegory.

Mike Shulman (Aug 23 2022 at 22:52):

Ah, I see.