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Stream: learning: reading & references

Topic: Semi-direct monoidal product on Set²

Vincent Moreau (Feb 08 2024 at 09:06):

Dear category theorists,

In the last few days, I found a cute construction. The category $\mathbf{Set}^2$ can be endowed with a monoidal structure given by the set-level semi-direct product

$$(Q, A) \rtimes (R, B) = (Q + (A \times R), A \times B)$$

whose unit is the object $(0, 1)$. Then, if I am not mistaken, the category of monoids in this monoidal category $\mathbf{Set}^2$ is isomorphic to the category of pairs $(Q, M)$ where $M$ is a (usual) monoid with a left action on the set $Q$.

Is this tensor product over $\mathbf{Set}^2$ already known, and if yes, what about the identification of its monoids?

PS: First time posting here!

Ralph Sarkis (Feb 08 2024 at 14:35):

Cute indeed. Is this operation still meaningful when considering only the skeleton of $\mathbf{FinSet}$? Maybe it will be easier to find references to that operation on $\mathbb{N}^2$.

Vincent Moreau (Feb 08 2024 at 15:37):

Indeed, this monoidal product restricts to $\mathbf{FinSet}$ and its monoids are then left actions of a finite monoid on a finite set! This is actually one of the reasons I consider it, in connection with automata theory.

Matteo Capucci (he/him) (Feb 09 2024 at 08:54):

I believe @Nima Motamed was telling me about this product once, in relation with semigroups if I'm not mistaken. Let's see if he joins in.

fosco (Feb 09 2024 at 09:36):

Vincent Moreau said:

Dear category theorists,

In the last few days, I found a cute construction. The category $\mathbf{Set}^2$ can be endowed with a monoidal structure given by the set-level semi-direct product

$$(Q, A) \rtimes (R, B) = (Q + (A \times R), A \times B)$$

whose unit is the object $(0, 1)$. Then, if I am not mistaken, the category of monoids in this monoidal category $\mathbf{Set}^2$ is isomorphic to the category of pairs $(Q, M)$ where $M$ is a (usual) monoid with a left action on the set $Q$.

Is this tensor product over $\mathbf{Set}^2$ already known, and if yes, what about the identification of its monoids?

PS: First time posting here!

In the extended version of "Differential 2-rigs", which unfortunately does not appear anywhere yet, @Todd Trimble and I give the following definition:

fosco (Feb 09 2024 at 09:36):

Let $\mathcal{M}$ be an $\mathcal{A}$-bimodule (i.e. an actegory for $\mathcal{A}$, with actions on both sides); one can define the square-zero extension of $\mathcal{M}$ to be the 2-rig $\mathcal{A}\ltimes \mathcal{M}$ whose underlying category is $\mathcal A \times \mathcal M$ and a 2-rig structure determined by

• componentwise coproducts (i.e. the sum $(A,M)\cup (B,N)$ is defined as $(A\cup B, M\cup N)$ for $A,B\in\mathcal{A}$ and $M,N\in\mathcal{M}$)

• multiplication defined as follows using the left and right action:

  $(A,M)\boxtimes(B,N) := (A\otimes B, A\lhd N \cup M\rhd B).$

fosco (Feb 09 2024 at 09:37):

Am I mistaken in thinking that whenever $\mathcal M$ is a Cartesian category, $\mathcal A=\mathcal M$ acts by "regular representation" (i.e. under the Cartesian product) on one side, and trivially on the other, this is your construction?

Vincent Moreau (Feb 09 2024 at 09:46):

Indeed, it seems to be exactly what you describe!

fosco (Feb 09 2024 at 09:57):

I'm happy to share ideas, let's hear from Todd (he might have thought about something else, or have another example in mind!)

Todd Trimble (Feb 09 2024 at 11:39):

Thanks, Fosco. No, this is pretty much what I might have said myself. I might have added that maps into this square zero extension correspond to monoidal functors $\mathcal{B} \to \mathcal{A}$ together with a functor $D: \mathcal{B} \to \mathcal{M}$ carrying a suitable Leibniz constraint $D(b \otimes b') \cong \phi(b) D(b') \cup D(b) \phi(b')$ where the right side refers to the actegory action. (There are some ideas in gestation about fruitful ways to extend this further, but this is good for now.)

Nima Motamed (Feb 15 2024 at 17:38):

Matteo Capucci (he/him) said:

I believe Nima Motamed was telling me about this product once, in relation with semigroups if I'm not mistaken. Let's see if he joins in.

Hiya! Thanks for the poke, I'm always very late with zulip.
Yeah indeed this construction came up in a discussion semimonads in Prof (which are very cute little gadgets). I've been putting off writing a preprint about this at some point, but basically, semimonads in Prof can be described as enriched categories in an appropriate way (and that generalizes also beyond Prof to enriched versions of it), and in trying to describe the base of enrichment there seemed to be some superficial resemblance to a semi-direct product just like this. I didn't manage to further develop that superficial resemblance, and instead found a different, 2-rig based description based on a categorified Dorroh extension. Reading what @fosco and @Todd Trimble write here, perhaps that superficial connection has something deeper going on after all? My curiosity is piqued.