Category Theory
Zulip Server
Archive

You're reading the public-facing archive of the Category Theory Zulip server.
To join the server you need an invite. Anybody can get an invite by contacting Matteo Capucci at name dot surname at gmail dot com.
For all things related to this archive refer to the same person.

Stream: learning: questions

Topic: G-monoids and G-algebras

Emily (she/her) (Feb 24 2025 at 13:56):

Relaying here a question I asked on MO, since maybe it's also of interest here:

The topic of this question is a little elementary, but it's something that I'm finding surprisingly hard to find good references about.

A while ago I was collecting examples of Picard and Brauer 2-groups of monoidal categories, and decided to try computing these for some categories.

• The first category I looked at was the category $\mathsf{Sets}_{G}$ of $G$-sets for a group (or monoid) $G$ together with the Cartesian product.
• Then, there was the category of $(G,G)$-sets with the balanced tensor product $X\boxtimes_{G}Y=X\times Y/(xg,y)\sim(x,gy)$.
• A different and similarly interesting category was $\mathrm{Rep}(G)$, the category of complex representations of a group $G$, equipped with either the usual tensor product or the $\mathbb{C}[G]$-tensor product, given by $V\boxtimes_{G}W=V\otimes_{\mathbb{C}}W/((\rho_g(v)\otimes w-v\otimes\rho_g(w))$.

Monoids in each of these categories seemed interesting:

1. Monoids in $\mathsf{Sets}_G$ with $\times$ are $G$-sets that are also monoids and we have $g\lhd(ab)=(g\lhd a)(g\lhd b)$.
2. Monoids in $\mathsf{Sets}_{(G,G)}$ with $\boxtimes_G$ are $(G,G)$-sets that are also monoids and satisfy $(a\rhd g)b=a(g\lhd b)$.
3. Monoids in $\mathsf{Rep}_G$ are representations $\rho\colon G\to\mathrm{GL}(V)$ of $G$ together with a product $V\otimes_{\mathbb{C}}V\to V$ on $V$ making it into a complex algebra and such that $\rho_{g}(vw)=\rho_g(v)\rho_g(w)$.
4. Monoids in $\mathsf{Rep}_G$ with $\boxtimes_G$ are similar, but satisfy $\rho_g(v)w=v\rho_g(w)$ instead.

Question. Are there references carrying a detailed study of the notions described above?

The third notion is the only one for which I was able to find satisfactory references, like this one or this one. Meanwhile, I've had a hard time finding anything written about the rest. Bourbaki's groups with operators come close to the first notion above, but are still a bit different.

Joe Moeller (Feb 24 2025 at 20:11):

If I was going to try to find sources for this, the first places I would look would be:

• Tensor categories by Etingof, Gelaki, Nikshych, Ostrik
• Monoidal Functors, Species, and Hopf Algebras by Mahajan and Aguiar
• Multiplication Objects in Monoidal Categories by Lopez

I don't know, but that's just where I would look.

John Baez (Feb 25 2025 at 03:22):

Monoids in $\mathsf{Set}_G$ would usually be called "actions of $G$ on monoids" or "$G$-equivariant monoids" or simply "$G$-monoids".

I went to MathSciNet and typed in "equivariant monoids" and got references to two papers. Here is one:

• Daniel Graves, Categorifying equivariant monoids, Proc. Amer. Math. Soc. 9 (2024), 3689–3704.

I'm afraid he's using "categorifying" in the sloppy sense of "throwing categories at" rather than "boosting up by replacing sets with categories". But that's okay for what Emily wants.

At first I hoped this paper would have many useful references, since the first sentence in the abstract is

Equivariant monoids are very important objects in many branches of mathematics: they combine the notion of multiplication and the concept of a group action.

However, it seems a bit thin on concrete examples, e.g. one of the "examples" is monoids in $\mathsf{Rep}_G$, which people would call "$G$-equivariant algebras" or simply "$G$-algebras". It has one reference to work on these, but there must actually be dozens.

For example if you gave me a group I could really sink my teeth into, like $\mathrm{U}(1)$ or $\mathrm{SU}(2)$, I could tell you shitloads of stuff about $G$-algebras.

John Baez (Feb 25 2025 at 03:26):

The second paper seems of dubious relevance so I won't go into it.

Emily (she/her) (Feb 28 2025 at 17:46):

Thanks Joe and John! Searching for "equivariant groups" led me to this nLab page and this book, which have the following proposition (here restated for monoids):

Let $A$ be a monoid. There is a fully faithful functor

$$\mathsf{Mon}(\mathsf{Sets}_{A},\times)\to\mathsf{Mon}_{/A}$$

from the category of monoids in $\mathsf{Sets}_A$ with respect to the Cartesian product [notion 1 above] to the category of monoids over $A$.

The functor is given by sending a left $A$-set $(B,\alpha\colon A\to\mathrm{End}(B))$ to the pair $(B\rtimes_{\alpha}A,p)$ with $B\rtimes_{\alpha}A$ the semidirect product monoid of $(A,B,\alpha)$ and $p\colon B\rtimes_{\alpha}A\to A$ the canonical projection.

I think with this the only notion that remains a bit mysterious to me is that of monoids in $\mathsf{Sets}_{(G,G)}$ with respect to the balanced tensor product, where we have a condition of the form $(a\rhd g)b=a(g\lhd b)$. Maybe there's also a rephrasing of it in terms of more commonly used notions...

John Baez (Feb 28 2025 at 18:02):

In the last equation are you using $a,b$ to mean elements of $G$ and $g$ to mean an element of the $(G,G)$-set?

I haven't thought about monoids in $(G,G)$-sets. I've thought about $(G,G)$-bitorsors: a $(G,G)$-bitorsor is a set which is both a left and right [[torsor]] for a group $G$ in a way obeying the equation you just wrote down.

Emily (she/her) (Feb 28 2025 at 19:12):

I've been a bit careless with writing the conditions down --- sorry for the confusion!

Emily (she/her) (Feb 28 2025 at 19:12):

What I mean is this: for $A$ a monoid, a monoid in $\mathsf{Sets}_{(A,A)}$ with respect to the balanced tensor product is a set $X$ with left and right actions

$$\begin{align*} \lhd_{A} &\colon A\times X \to X,\\ \rhd_{A} &\colon X\times A \to X \end{align*}$$

of $A$ along with a monoid structure on $A$ such that we have

$$(x\rhd_{A}a)y = x(a\lhd_{A}y)$$

for each $x,y\in X$ and each $a\in A$. In addition, the multiplication map should also be equivariant, so we should also have

$$\begin{align*} a\lhd_{A}(xy) &= (a\lhd_{A}x)y,\\ (xy)\rhd_{A}a &= x(y\rhd_{A}a),\\ \end{align*}$$

for each $a\in A$ and each $x,y\in X$ (I missed these conditions earlier).

Emily (she/her) (Feb 28 2025 at 20:31):

I think I got it! $(A,A)$-sets are the same as profunctors on the one-object category associated to $A$, and in this case the balanced tensor product corresponds to profunctor composition, so these are the same as promonads on $A$, i.e. identity on objects functors from that one-object category to another category. So we are looking at maps of monoids $A\to B$. This gives an equivalence of categories

$$\mathsf{Mon}(\mathsf{Sets}_{(A,A)},\boxtimes_{A}) \cong \mathsf{Mon}_{A/},$$

surprisingly enough.

John Baez (Feb 28 2025 at 20:55):

Emily (she/her) said:

I think I got it! $(A,A)$-sets are the same as profunctors on the one-object category associated to $A$, and in this case the balanced tensor product corresponds to profunctor composition, so these are the same as promonads on $A$, i.e. identity on objects functors from that one-object category to another category.

I hope "these" means not "$(A,A)$-sets" but "monoids internal to $(A,A)$-sets". (As a pedant, I hate pronouns whose antecedents are unclear to me.) I believe from what you said that

• $(A,A)$-sets are the same as profunctors from the one-object category $BA$ to itself
• monoids internal to $(A,A)$-sets are the same as [[promonads]] on $BA$.

Does that sound right?

If so, cool! I hadn't known that promonads on a category are the same as identity-on-objects functors from that category to some other, but after a few minutes that's obvious too. Cool!

Emily (she/her) (Mar 01 2025 at 18:50):

I hope "these" means not "$(A,A)$-sets" but "monoids internal to $(A,A)$-sets". (As a pedant, I hate pronouns whose antecedents are unclear to me.) I believe from what you said that

Yep! (And you're absolutely right — I'll be careful to avoid this issue in my future writing!)

Emily (she/her) (Mar 01 2025 at 18:51):

Does that sound right?

Yep!

Emily (she/her) (Mar 01 2025 at 18:53):

By the way, later yesterday I was talking about this result with a friend and colleague (Violeta Martins de Freitas) and she mentioned something that made this result much more clear to me: this is the $G$-set version of the bijection between $R$-algebras viewed as monoids in $R$-modules vs. as rings $A$ with a map $R\to A$

John Baez (Mar 01 2025 at 18:55):

Great! Once I mentally process your chain of reasoning, it becomes "visually obvious in examples" that a monoid in $(A,A)$-sets is an identity-on-objects functor from $BA$ to some other category. So this is really beautiful.

And note that there's an even simpler name for an "identity-on-objects functor from $BA$ to some other category". It's a monoid homomorphism from $A$ to some other monoid! Oh, you already said that. :sweat_smile:

Emily (she/her) (Mar 01 2025 at 19:00):

And note that there's an even simpler name for an "identity-on-objects functor from $BA$ to some other category". It's a monoid homomorphism from $A$ to some other monoid!

Yep, exactly!

Emily (she/her) (Mar 01 2025 at 19:00):

I also find it funny that the reason I figured this was by thinking about promonads — usually it'd be the other way around, like trying to come up with examples of promonads and figuring what they in the case of one-object categories

Emily (she/her) (Mar 01 2025 at 19:01):

Oh, you already said that. :sweat_smile:

To be fair I do find it a cool enough realization for it to be worth to be said twice! :)