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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
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