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I’m looking at these “actegories” because I noticed that models of call-by-push-value can be described as actions of a (cartesian) monoidal category on a category with a dual adjunction between the two categories compatible with the action. As an aside, I don’t like the name actegory and would just prefer action of a monoidal category.
Searching for applications for CS/programming I’m curious if there are any common uses of actions of a monoidal category?
The only idea I have is that as monoidal categories are the “right” place to define monoids, an action of a monoidal category is the “right” place to define the action of a monoid: If C acts on D you can define a monoid in C acting on an object of D. Does anyone have any examples where things are fruitfully framed this way?
I don’t like the word “actegory” either.
One example of this “right place to define monoids” thing is the definition of the bar construction. If $V$ is a monoidal category acting on $C$, and $D\in V$ is a monoid and $M\in C$ is a $D$-module, then there is a simplicial object $B_\bullet(D,D,M) \in C^{\Delta^{op}}$ where $B_n(D,D,M) = D^{\otimes (n+1)} \odot M$, whose realization is a “resolution” of $M$ qua $D$-module. At this level the gain is perhaps not obvious, since this bar construction factors through the “universal” action of $End(C)$ on $C$, with $D$ corresponding to the monad (= monoid in $End(C)$) $T_D(X) \coloneqq D\odot X$. But when you talk about two-sided bar constructions $B(N,D,M)$ and “coend-like” bar constructions $B(D,H)$ I think the extra generality can be useful. Although it’s been a while since I thought about this sort of thing much.
Hi – The Kelly-Janelidze paper has a nice idea: an enriched category with copowers is the same thing as an actegory that has a right adjoint in the appropriate argument. This leads to a simpler or even more “natural” presentation for enriched categories with copowers. If you are looking for examples, Rasmus Mogelberg and I used it extensively in linear usage of state and I found it useful in Freyd categories are enriched Lawvere theories.
There is also a really nice characterization of enrichment with finite copowers in a locally D-presentable smc, in terms of actions (Prop 2.7 in my Freyd cats paper, but see also “Enrichment through variation”).
Plenty of other people have also found this kind of thing useful including Marcelo Fiore, Paul Levy and Paul-Andre Mellies. This might all be related to call-by-push-value.
Actions of monoidal categories are very important in geometric representation theory. For example, everything under the name 2-representation theory in the sense of Rouquier is about this: $sl_2$-categorification, 2-Kac-Moody algebras, etc.
Thanks everyone for the examples. Unsurprisingly Sam’s are the most understandable to me at the moment. I came to my current model of CBPV from the Enriched Effect Calculus work which I described somewhat here: https://ncatlab.org/nlab/show/call-by-push-value#as_an_adjoint_logic . I sent you an email with more details!
A reference to add
Does anyone know where the term “actegory” originated? The earliest article I found with a little searching is the 2000 paper of McCrudden Categories of Representations of Coalgebroids.
In his Balanced Coalgebroids of the same year he writes
Indeed, Pareigis [Par96] uses the theory of V-actegories to facilitate reconstruction,
but the term isn’t used explicitly there.
The paper in #13 is described as the sequel to yours in #12, and both emerge from McCrudden’s PhD thesis “Categories of representations of balanced coalgebroids”, so perhaps there for the first appearance.
J. R. B. Cockett, Craig Pastro, The logic of message-passing:
The term actegory is used to describe the situation of a monoidal category “acting” on a category. They first appeared (under a different name) in the work of Bénabou as a simple example of a bicategory. B. Pareigis developed the theory of actegories (again under a different name) and showed there usefulness in the representation theory of monoids and comonoids. The word “actegory” was first suggested at the Australian Category Seminar and first appeared in print in the thesis of P. McCrudden [20] where they were used to study categories of representations of coalgebroids.
Thanks for investigating!
I have touched the Idea-section:
added more hyperlinkung, throughout
mentioned coherent internalization of action objects in $Cat$
mentioned that actions of 2-ring-like categories would (rather) be called “2-modules”.
$\,$
(Curiously, it is category theorists, of all people, who are happy to use “module” for actions of non-linear monoids, so that they should have been happy to say something like “2-module” for categorified monoid actions.
I find the term “actegory” a remarkably unfit choice of terminology. When I was a student I used to think it was meant as a bad joke, until a colleague at a conference insisted to use it in his talk title.)
I find the term “actegory” a remarkably unfit choice of terminology. When I was a student I used to think it was meant as a bad joke,
Actually, now that I see #2 and #4 above: We should just go ahead an rename this entry to “action of a monoidal category” and relegate the “actegory”-pun to a historical side-remark.
19: when I first heard it, from Schaunburg’s talk in 2001, I did not like it either but by now it is quite standard terminology in fact rather than historical “pun”. It is indeed used widely by Pareigis’ Munich school and Hopf algebraists who used a lot of Hopf machinery in late 1990s at least. but there are also works in Australian category theory school as early as that. Published version of McCrudden’s article is from 2000:
and R. Garner recently (2018)
and Street as of 2011 https://arxiv.org/abs/1111.5659 and 2019 https://arxiv.org/abs/1903.03890.
Of course we should distinguish space acted upon and the action itself. Module or module category is used by some schools indeed, but it is even in categorical algebra overloaded term (some other versions of categorical modules are used as well). I don’t know, it is tricky. I think that people motivated by representation theory and CFT like Ostrik, Etingof etc. by module category mean additiive case with action biadditve and exact in each argument (see https://arxiv.org/abs/math/0111139 for example). So it is analogue of a linear module over a ring rather than space acted upon nonlinearly by a monoid or group.
P.S. In 1970s Pareigis has used the term $V-category$ rather than $V-actegory$ but people in his school switched to actegories in late 1990s or in 2000 at least. Notably Schauenburg from that school wrote several works starting from about that time. Weather they borrowed int from McCrudden I do not know, but regarding that the topic was Hopf algebraic (coalgebroids) it may be.
I am taking the liberty of doing what is right instead of what is common:
renamed the entry from “actegory” to “action of a monoidal category”.
have re-written the Idea-section (there seemed to be room for improvement even independently of the choice of terminology)
replaced all occurrences of “actegory” by “module category”
except in one paragraph where the alternative terminology is discussed
which now has the previous quote from Cocket & Pastro 2007 as a footnote.
Looks like we then need to change that Related Concepts section on the bimodule equivalent called there ’biactegory’, and so then the entry biactegory.
So I guess the latter becomes biaction of monoidal categories.
No, it should really just be “bimodule category”.
Re #23, well according to this current page, we’re only supposed to speak of ’module categories’
if some linear structure is present and respected.
Do people think of more general biactions?
according to this current page
You mean according to what I just wrote there. But I was careful to start the sentence you partially quote with “At least if..”
Because, as voiced in #18, I have a hard time seeing why in a context of hardcore category theory — where it is tradition to call any profunctor a “bimodule” — anyone should bat an eye over using “module category” for the most general notion of categorical action.
No, it should really just be “bimodule category”.
“Action” and “module” are generally synonymous in category theory (though in some contexts, some terms have historically been preferred, and in some either is used). I see no reason why “bimodule” should be preferred over “biaction”, especially when “action” rather than “module” has historically been used in the context of monoids.
I am taking the liberty of doing what is right instead of what is common:
I personally find it a bit objectionable to write:
Beware that, alternatively, an old pun promoted in McCrudden (2000) is still fairly widely used by some authors, who find it reasonable to speak of “actegories”1 (and then, for better or worse, “biactegories” for the two-sided case).
Elsewhere, standard terminology has been promoted on the nLab, and alternative terminology mentioned without prejudice, except where there is good reason for it. This comments feels pejorative without, I feel, good reason. You may dislike the pun, but that doesn’t mean that it’s not reasonable, or that there aren’t people who prefer the terminology. I would prefer to reword this line.
Please feel free to adjust the wording where you feel it’s objectionable. (I am not sure which word you are objecting to, but just go ahead)
I would be fine with “biaction-category” or similar. But “biactegory” seems cringy and needlessly so.
I’ve made an adjustment to the sentence, in particular giving an objective reason to avoid the term “actegory”. I may have picked up on an implication that wasn’t intended in the previous phrasing, but I feel in any case that it is better to be overly careful when presenting preferred alternatives to usual terminology.
29 biaction category would be indeed a good choice.
31: Though I like it I am partially taking it back: as you know, there is a notion of action category for a category or more often for a group(oid) action (no monoidal categories in the game). Baković also has its categorification, an action bicategory for bicategorical action.
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