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Stream: theory: category theory

Topic: Hopf monoids in set species

Simon Burton (Nov 26 2023 at 15:44):

In the paper "Hopf monoids and generalized permutahedra" by Aguiar and Ardila, (also discussed here) they define a connected Hopf monoid in set species:
image.png

Simon Burton (Nov 26 2023 at 15:45):

That's a nice, concrete description, that looks like it should have a slick categorical version. I'm thinking of something like "A Hopf monoid in set species is a (pseudo?) Hopf monoid object in the (bi?)category of ...", and i'm wondering what is the "..." First of all I see something that looks like the conditions for a lax and oplax monoidal functor, and the nlab even uses the same letters $\mu$ and $\sigma$, which makes me wonder if this Hopf monoid in set species construction is well known to the category theorists, possibly in some other contexts.

Simon Burton (Nov 26 2023 at 16:31):

I guess I'm hoping to see the string diagram for the bialgebra rule
image.png
done with laxators and colaxators (is that the right terminology?) $\mu,\sigma$.

Simon Burton (Nov 26 2023 at 16:37):

Now i'm thinking of a monoidal category as a pseudomonoid (in the cartesian bicategory of categories). These pseudomonoids have string diagrams that look like this
image.png

Simon Burton (Nov 26 2023 at 16:38):

where this is a sheet for the underlying category, and the monoidal product merging two sheets into one (i use right-to-left direction) and the unit.

Simon Burton (Nov 26 2023 at 16:39):

there's also associator and unitor 2-cells that look like this
image.png

Simon Burton (Nov 26 2023 at 16:41):

that satisfy various equations.. Now if i take another monoidal category
image.png

Simon Burton (Nov 26 2023 at 16:42):

then a lax monoidal functor from the grey to the blue monoidal categories will have these pieces
image.png

Simon Burton (Nov 26 2023 at 16:53):

And a colax monoidal functor would be the same diagrams reflected vertically
image.png

Simon Burton (Nov 26 2023 at 16:55):

and we would require the lax and colax structure to be on the same functor. (I hope i didn't mix anything up so far, let me know if i already went off the rails..)

Simon Burton (Nov 26 2023 at 16:59):

So my first attempt at the left-hand side of the bialgebra rule looks like this
image.png

Simon Burton (Nov 26 2023 at 17:01):

and this is promising, but there's not enough grey sheets coming out of this. As we pull the middle of the blue line over to the left we want it to bifurcate onto four blue lines (four copies of the functor), so this is going to get a bit complicated.

Simon Burton (Nov 26 2023 at 17:20):

Something more like this
image.png

John Baez (Nov 26 2023 at 19:54):

Simon Burton said:

That's a nice, concrete description, that looks like it should have a slick categorical version. I'm thinking of something like "A Hopf monoid in set species is a (pseudo?) Hopf monoid object in the (bi?)category of ...", and i'm wondering what is the "..." First of all I see something that looks like the conditions for a lax and oplax monoidal functor, and the nlab even uses the same letters $\mu$ and $\sigma$, which makes me wonder if this Hopf monoid in set species construction is well known to the category theorists, possibly in some other contexts.

Set species are just a symmetric monoidal category, not bicategory, so I don't think "pseudo" does anything for us here. Set species are the free cocomplete 2-rig on one object and my impression was that those guys were looking at Hopf monoids in there... I should check. Actually aren't they using Vect-valued species?

Christian Williams (Nov 27 2023 at 00:21):

These images are great! What tool did you use?

Christian Williams (Nov 27 2023 at 00:38):

Is that last drawing the LHS of the bialgebra rule? Why does it involve a braiding (in the grey sheets)?

Simon Burton (Nov 27 2023 at 05:01):

John Baez said:

Set species are just a symmetric monoidal category, not bicategory, so I don't think "pseudo" does anything for us here.

Simon Burton (Nov 27 2023 at 05:01):

A pseudo bimonoid (or pseudo hopf monoid) would involve a pseudomonoid
image.png

Simon Burton (Nov 27 2023 at 05:02):

(right to left order)
and a pseudo comonoid
image.png

Simon Burton (Nov 27 2023 at 05:05):

(as well as units, associators, etc.) and then the pseudo bialgebrator would look like this:
image.png

Simon Burton (Nov 27 2023 at 05:07):

So this is not what i drew yesterday, but it does involve the pseudo direction, at least diagonally.

Simon Burton (Nov 27 2023 at 05:39):

Christian Williams said:

These images are great! What tool did you use?

Yeah i find string diagrams extremely helpful, pretty much a revolution in my ability to do calculations. This tool is called wiggle.py. I gave a talk about it at SYCO 11.

Simon Burton (Nov 27 2023 at 05:41):

Christian Williams said:

Is that last drawing the LHS of the bialgebra rule? Why does it involve a braiding (in the grey sheets)?

Well, i don't think this is right yet, but the idea was to drag the blue region over to the right
image.png

Simon Burton (Nov 27 2023 at 05:42):

that's the bottom layer at least, involving two copied of the oplaxators. Then the middle layer would look like this:
image.png

Simon Burton (Nov 27 2023 at 05:46):

so that's at least starting to look like the right hand side of the bialgebra rule,
image.png

Simon Burton (Nov 27 2023 at 05:47):

looking into the "maw" of the previous diagrams, without the shading.

Simon Burton (Nov 27 2023 at 05:50):

But you can see it's not going to work when i try to compose the laxators above this. So somehow I need to do something a bit different with the symmetry 1-cells
image.png

Simon Burton (Nov 27 2023 at 05:50):

image.png

Simon Burton (Nov 27 2023 at 05:56):

These are symmetric pseudomonoids, which means we have symmetry 2-cells
image.png

Simon Burton (Nov 27 2023 at 05:57):

etc. My guess is that these are involved as well.

Simon Burton (Nov 27 2023 at 06:01):

John Baez said:

Set species are the free cocomplete 2-rig on one object and my impression was that those guys were looking at Hopf monoids in there... I should check. Actually aren't they using Vect-valued species?

Ok. They define Vect valued species in section 2.3. But they start with set valued species in section 2.1 and hopf monoids therein.

John Baez (Nov 27 2023 at 09:47):

Okay. I have a vague memory that most of their examples of Hopf monoids are in Vect-valued species, and I thought there was a good reason why but now I don't see one. So maybe they do have some nice Hopf monoids in Set-valued species.

John Baez (Nov 27 2023 at 09:48):

In a cartesian category every object is a comonoid in exactly one way, so Hopf monoids are sort of limited as far as their comonoid structure goes.

John Baez (Nov 27 2023 at 09:51):

But when they look at Hopf monoids in Set-valued species, they are (I'm pretty sure) not using the cartesian monoidal category structure on Set-valued species: they're using the "Day convolution" monoidal category structure - the most famous way to multiply species.

Simon Burton (Nov 27 2023 at 10:11):

Thankyou. I guess it's homework for me to see if these Day-convolution Hopf monoids correspond to what I was saying above about lax and oplax structures interacting, and also to check how this relates to the definition in the paper itself.

John Baez (Nov 27 2023 at 13:47):

If you find an example of a Day-convolution Hopf monoid in Set-valued species let me know! I should probably know some but I'm drawing a blank.

Simon Burton (Nov 27 2023 at 14:16):

There's Example 2.8 in the paper. The underlying species is the species of linear orders, the product is concatenation and the coproduct is splitting .... is this fitting with the Day-convolution picture? ... i think it might be.
image.png

John Baez (Nov 27 2023 at 22:14):

Can you tell me which paper that is and where is Example 2.8? I'm looking at their huge book Monoidal Functors, Species and Hopf Algebras and not finding that, so maybe you're reading something else.

John Baez (Nov 27 2023 at 22:15):

Oh, okay: Hopf monoids and generalized permutahedra by Aguiar and Ardila!

John Baez (Nov 27 2023 at 22:15):

Btw, that huge book is really good if you're interested in Hopf monoids in species.

John Baez (Nov 27 2023 at 22:18):

Looking at Def. 2.5 in Aguiar and Ardila's paper, it seems they are indeed defining a Hopf monoid in the monoidal category of species where the monoidal structure is Day convolution with respect to the "disjoint union" monoidal structure on the groupoid of finite sets.

John Baez (Nov 27 2023 at 22:20):

I think I see how the species of linear orders is a bimonoid in this way. The multiplication takes a linear ordering on the finite set $S$ and a linear ordering on the finite set $T$ and produces a linear ordering on $S \sqcup T$ in the obvious (highly noncommutative way).

John Baez (Nov 27 2023 at 22:24):

The comultiplication takes a linear ordering on a finite set $S$ and a decomposition $S = T \sqcup T'$ and gives linear orderings on $T$ and $T'$, again in an obvious way.

John Baez (Nov 27 2023 at 22:25):

But what about the antipode? The funny thing is that their definition of Hopf monoid, or more precisely "connected Hopf monoid", seems to be a definition of what I'd call a bimonoid! I don't see them mentioning the antipode!

John Baez (Nov 27 2023 at 22:26):

Maybe I'm missing it. But if they are really talking about bimonoids, not what I'd call Hopf monoids, then it's much easier for me to imagine examples.

Simon Burton (Nov 28 2023 at 05:54):

Yeah they don't have an antipode on their Hopf monoids in set species.

John Baez (Nov 28 2023 at 08:35):

So I think they just decided that "Hopf algebra" sounds sexier than "bialgebra".

Mike Shulman (Nov 28 2023 at 09:51):

Using "Hopf algebra" to mean "bialgebra" is unfortunately fairly common in some circles. One reason for this in algebraic topology is that a graded bialgebra that is "connected" in the sense that its 0-degree part is the ground ring is automatically a Hopf algebra, and these are the sorts that arise as the (co)homology of connected spaces. We had a bit of a conversation about this a few years ago... (-:

Simon Burton (Jun 24 2024 at 17:19):

I'm still hoping to find the appropriate string diagrams for this, and have made an epsilon of progress.

Simon Burton (Jun 24 2024 at 17:22):

Thanks to @Aaron David Fairbanks for pointing out that the above diagrams for monoidal functors also appear in this work by Simon Willerton. Although i don't think the Hopf monads there are the same as these Hopf monoids.

Simon Burton (Jun 24 2024 at 17:29):

Just to recap (see here, section 2), we have two monoidal categories:
$Set_\times$ which is sets with cartesian product, and
$Bij_{+}$ finite sets, bijections, and disjoint union.
Then a set species is a functor $H$ from $Bij$ to $Set$. A Hopf monoid in set species is a set species $H$ that is furthermore lax monoidal, and oplax monoidal and hopf.

Simon Burton (Jun 24 2024 at 17:32):

I'm reasonably sure this is correct so far, but it's this last "hopf" condition that i'm trying to figure out, and in particular turn into a string diagram.

Simon Burton (Jun 24 2024 at 17:36):

Here is the hopf condition from Aguiar and Ardila:
image.png

Simon Burton (Jun 24 2024 at 17:41):

There is a nice string diagram in $Bij_{+}$ which at least captures the hypothesis of this condition
image.png

John Baez (Jun 24 2024 at 19:47):

That string diagram is the main law governing bialgebras - or I'd prefer to say 'bimonoids'. It shows up when you have a monoid that's also a comonoid, and it says

• the monoid multiplication is a comonoid homomorphism

Equivalently, it says

• the comonoid comultiplication is a monoid homomorphism.

So there's nothing especially 'Hopf' about this, since a Hopf algebra is a bialgebra with an extra unary operation called the 'antipode', obeying some laws of its own.

To illustrate the difference:

Whenever $M$ is a monoid and $k$ is a field, $k[M]$ (the set of $k$-linear combinations of elements of $M$) is a bialgebra.

Whenever $G$ is a group $k$ is a field, $k[G]$ (the set of $k$-linear combinations of elements of $G$) is a Hopf monoid.

The inverse operation on the group gives $k[G]$ its antipode.

John Baez (Jun 24 2024 at 19:48):

More generally we can define bimonoids and Hopf monoids in any symmetric monoidal category. There are people who don't strongly distinguish betweem bimonoids and Hopf monoids, but I think that's a bit sad because they're both interesting concepts, which work differently (though every Hopf monoid is a bimonoid).

Simon Burton (Jun 25 2024 at 09:15):

Thanks for going over that terminological confusion again.

So I seem to be asking for the definition of a lax & oplax symmetric monoidal functor, whose laxator & oplaxator obey a bimonoid rule. And indeed, this is not to hard to write down, once i got over trying to string-diagram it. (Maybe functor boxes would work better for this, btw). After pondering what such a thing should be called, it seems like "bilax monoidal functor" is right. And lo! there is such a page on the nlab. This is exactly the definition I had just written down in my notebook here. Which leads to another obvious question, what is the a Frobenius version of this?

John Baez (Jun 25 2024 at 09:24):

Since a lot of interesting Hopf algebras are also Frobenius algebras, if you're studying Hopf monoids or bimonoids in species, you should definitely take a look at Frobenius monoids in species - i.e., Frobenius monoidal functors from the groupoid of finite sets with the disjoint union monoidal structure to $(\mathsf{Set}, \times)$.

John Baez (Jun 25 2024 at 09:28):

So Aguiar and Mahajan, or Aguiar and Ardila, haven't done that?

Simon Burton (Jun 25 2024 at 09:36):

Yes I'm just reading Aguiar and Mahajan now, and this is where I should have started.. doh. They show that a bilax monoidal functor that is "normal" satisfies the Frobenius identites (Proposition 3.41.)

Noam Zeilberger (Jun 26 2024 at 15:36):

Hi there @Simon Burton! So I thought a bit about this question some years ago and convinced myself that there was a natural fibrational interpretation of "Hopf monoids in set species" as pseudo bimonoids over the category FinBij of finite sets and bijections, viewing the latter as a pseudo bimonoid in Prof. It's been a while but let me try to explain the idea.

As with any monoidal category, we can view the monoidal category of finite sets and bijections (FinBij,+) as a pseudo monoid in (Cat,×), which is then automatically transported to both a (pseudo) monoid and a comonoid in Prof (the bicategory of small categories, profunctors, and natural transformations) by the covariant and contravariant embedding functors Cat → Prof and Catᵒ → Prof. In general, a monoidal category only induces a pseudo monoid and a pseudo comonoid in Prof, which cannot necessarily be combined into a bimonoid. However, it so happens that the induced monoid and comonoid structure on FinBij (i.e., the adjoint pair of profunctors + : FinBij × FinBij ↛ FinBij and +' : FinBij ↛ FinBij × FinBij corresponding to the disjoint union functor + : FinBij × FinBij → FinBij) can be equipped with the extra structure turning it into a bimonoid in Prof. (Probably for good reasons, but I never thought hard about them beyond asking a related question on Mathoverflow.)

Now, of course a species is defined as a presheaf on FinBij, which can equivalently be interpreted as a profunctor 1 ↛ FinBij. In turn, by taking the lax slice under 1, we can view such species-as-profunctors as objects of a bicategory Prof• := 1 // Prof whose objects are pairs (C,S) of a category C and a profunctor S : 1 ↛ C, and whose 1-cells (C,S) → (D,T) are pairs (M,θ) of a profunctor M : C ↛ D and a natural transformation θ : M ∘ S ⇒ T, with 2-cells defined as expected. This bicategory Prof• can be equipped with a monoidal structure refining the monoidal structure of Prof, where the tensor product of two presheaves (C,S) ⊗ (D,T) is the presheaf (C×D, S⊗T) defined by (S⊗T)(c,d) = S(c) × T(d). Paul-André Melliès and I discussed this construction in A bifibrational reconstruction of Lawvere's presheaf hyperdoctrine; the forgetful functor Prof• → Prof is actually an example of a monoidal closed bifibration, in the sense that the functor is both a bifibration and strict monoidal closed.

So now the point is that a "Hopf monoid in set species" can be defined as a pseudo bimonoid in Prof• lying over the pseudo bimonoid (FinBij,+,+') in Prof. In other words, a species S equipped with a pair of arrows μ : (FinBij,S) ⊗ (FinBij,S) → (FinBij,S) and Δ : (FinBij,S) → (FinBij,S) ⊗ (FinBij,S) in Prof• lying over the profunctors + : FinBij × FinBij ↛ FinBij and +' : FinBij ↛ FinBij × FinBij, satisfying the bimonoid laws up to iso. If you work out what that means, you recover exactly the definition in Aguilar & Mahajan / Aguilar & Ardila, at least as I remember it. Crucially, here we are making use of the "external" tensor product S ⊗ S in Prof• which defines a presheaf over the product category FinBij × FinBij, rather than the Day convolution of species. I vaguely remember reading a comment in the Aguilar & Mahajan book (or perhaps it was another paper of Aguilar?) where they explicitly describe/lament the fact that the underlying comultiplication structure of a Hopf monoid in set species does not give a comonoid in the monoidal category of set species with the Day convolution product. And this was my explanation for why, although I did not try to take it any further.

Noam Zeilberger (Jun 26 2024 at 19:44):

I think the remark I had in mind was on p.23 of Aguilar and Mahajan's Hopf monoids in the category of species:

The situation for comonoids is different. The existence of the counit forces a
comonoid Q in (Sp, ·) to be concentrated on the empty set. Indeed, a morphism
ε : Q → 1 entails maps εI : Q[I] → 1[I], and hence we must have Q[I] = ∅ for all
nonempty I.

There is, nevertheless, a meaningful notion of set-theoretic comonoid. It consists, by definition, of a set species Q together with a collection of maps ∆S,T : Q[I] → Q[S] × Q[T ], one for each I = S ⊔ T , subject to the coassociative and counit axioms.

As I wrote above, the way I understand this definition is that they are really asking for a comonoid in Prof• lying over the comonoid (FinBij,+') in Prof.