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Stream: learning: questions

Topic: G-monoids and G-algebras


view this post on Zulip 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.

Monoids in each of these categories seemed interesting:

  1. Monoids in SetsG\mathsf{Sets}_G with ×\times are GG-sets that are also monoids and we have g(ab)=(ga)(gb)g\lhd(ab)=(g\lhd a)(g\lhd b).
  2. Monoids in Sets(G,G)\mathsf{Sets}_{(G,G)} with G\boxtimes_G are (G,G)(G,G)-sets that are also monoids and satisfy (ag)b=a(gb)(a\rhd g)b=a(g\lhd b).
  3. Monoids in RepG\mathsf{Rep}_G are representations ρ ⁣:GGL(V)\rho\colon G\to\mathrm{GL}(V) of GG together with a product VCVVV\otimes_{\mathbb{C}}V\to V on VV making it into a complex algebra and such that ρg(vw)=ρg(v)ρg(w)\rho_{g}(vw)=\rho_g(v)\rho_g(w).
  4. Monoids in RepG\mathsf{Rep}_G with G\boxtimes_G are similar, but satisfy ρg(v)w=vρg(w)\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.

view this post on Zulip 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:

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

view this post on Zulip John Baez (Feb 25 2025 at 03:22):

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

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

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 RepG\mathsf{Rep}_G, which people would call "GG-equivariant algebras" or simply "GG-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 U(1)\mathrm{U}(1) or SU(2)\mathrm{SU}(2), I could tell you shitloads of stuff about GG-algebras.

view this post on Zulip John Baez (Feb 25 2025 at 03:26):

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

view this post on Zulip 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 AA be a monoid. There is a fully faithful functor

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

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

The functor is given by sending a left AA-set (B,α ⁣:AEnd(B))(B,\alpha\colon A\to\mathrm{End}(B)) to the pair (BαA,p)(B\rtimes_{\alpha}A,p) with BαAB\rtimes_{\alpha}A the semidirect product monoid of (A,B,α)(A,B,\alpha) and p ⁣:BαAAp\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 Sets(G,G)\mathsf{Sets}_{(G,G)} with respect to the balanced tensor product, where we have a condition of the form (ag)b=a(gb)(a\rhd g)b=a(g\lhd b). Maybe there's also a rephrasing of it in terms of more commonly used notions...

view this post on Zulip John Baez (Feb 28 2025 at 18:02):

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

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

view this post on Zulip Emily (she/her) (Feb 28 2025 at 19:12):

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

view this post on Zulip Emily (she/her) (Feb 28 2025 at 19:12):

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

A ⁣:A×XX,A ⁣:X×AX\begin{align*} \lhd_{A} &\colon A\times X \to X,\\ \rhd_{A} &\colon X\times A \to X \end{align*}

of AA along with a monoid structure on AA such that we have

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

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

aA(xy)=(aAx)y,(xy)Aa=x(yAa),\begin{align*} a\lhd_{A}(xy) &= (a\lhd_{A}x)y,\\ (xy)\rhd_{A}a &= x(y\rhd_{A}a),\\ \end{align*}

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

view this post on Zulip Emily (she/her) (Feb 28 2025 at 20:31):

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

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

surprisingly enough.

view this post on Zulip John Baez (Feb 28 2025 at 20:55):

Emily (she/her) said:

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

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

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!

view this post on Zulip Emily (she/her) (Mar 01 2025 at 18:50):

I hope "these" means not "(A,A)(A,A)-sets" but "monoids internal to (A,A)(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!)

view this post on Zulip Emily (she/her) (Mar 01 2025 at 18:51):

Does that sound right?

Yep!

view this post on Zulip 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 GG-set version of the bijection between RR-algebras viewed as monoids in RR-modules vs. as rings AA with a map RAR\to A

view this post on Zulip 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)(A,A)-sets is an identity-on-objects functor from BABA 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 BABA to some other category". It's a monoid homomorphism from AA to some other monoid! Oh, you already said that. :sweat_smile:

view this post on Zulip 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 BABA to some other category". It's a monoid homomorphism from AA to some other monoid!

Yep, exactly!

view this post on Zulip 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

view this post on Zulip 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! :)