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Stream: community: our work

Topic: Nathanael Arkor

Nathanael Arkor (Apr 01 2025 at 15:45):

I'm going to be giving a talk online at the Topos Colloqium this week at 2025-04-03T20:00:00+03:00 – everyone is very welcome to come along! I won't be assuming any prior familiarity with double categories or polynomials/polynomial functors.

Title: A (virtual) double category theorist's perspective on polynomials
Abstract:
I shall tell the tale of how I came to understand polynomials. In doing so, I will exhibit a novel universal property of the double category of polynomials in a locally cartesian closed category. In fact, we shall see how polynomials may be liberated from the assumption of exponentiability entirely. This talk is based on joint work with Bryce Clarke.

Link: https://youtu.be/tCbRfjv6JQ4

Nathanael Arkor (Apr 03 2025 at 16:01):

(The talk will be starting in an hour.)

David Michael Roberts (Apr 04 2025 at 02:58):

I really enjoyed this talk, I'd be interesting in knowing what your non-LCC Poly construction does for (infinitary) pretoposes (say with param. NNO), if anything. You need a Pi-pretopos for the traditional setting of polynomials, but this is sometimes just not possible. The internal logic is rather strong, and you can even get limited instances of decidability.

There are some polynomials functors in the traditional sense that do exist, for instance where the middle, right-pointing morphism in the polynomial is a something like a family of numerals. You can write down the polynomial $1 \leftarrow A \to B \to 1$ for the case of a 'naive polynomial endofunctor $X \mapsto \sum_{n=0}^\infty A_n \times X^n$ for a family of objects $A_n$ (at least, last I checked, a couple of years ago!). Here the morphism $A\to B$ is such that pullback along it does admit a right adjoint, because it only needs finite limits to construct. So you hopefully can embed the "polynomials that give polynomial functors" into your more general polynomials, in a reasonable way.

David Corfield (Apr 04 2025 at 07:54):

Interesting! That sense that a good way to view polynomials is by seeing what needs to be added to spans:
image.png
also occurs in Strickland's Tambara functors.

Mackey functors are to spans are to semigroups as Tambara functors are to polynomials are to semirings.

He calls polynomials 'bispans':
image.png

I wonder if anyone's looking at double categories of Mackey/Tambara functors.

David Corfield (Apr 04 2025 at 08:14):

Ross Street in Objective Mackey and Tambara functors via parametrized categories [arXiv:2503.02260] looks to take polynomials as iterated spans:

we can replace the original parametrizing base for objective Mackey functors by a bicategory of spans while the replacement for objective Tambara functors is a bicategory obtained by iterating the span construction; these iterated spans are polynomials.

Nathanael Arkor (Apr 04 2025 at 09:07):

@David Michael Roberts: I didn't have time to mention it during the talk, but there is a slight refinement of the covirtual double category of polynomials mentioned at the end of the talk, where we specify a class of "powerful" morphisms in our category $\mathcal E$, and require that the loose morphisms in the covirtual double category are the polynomials whose middle morphism is powerful.

If we take "powerful = exponentiable", then we obtain a double category of polynomials in $\mathcal E$ where $\mathcal E$ is not required to be locally cartesian closed. (This extends the bicategory of polynomials in Weber's paper Polynomials in categories with pullbacks.)

This modified covirtual double category admits a similar universal property to the one mentioned, and in particular gives an interpretation of polynomials (with middle morphism exponentiable) as polynomial functors in categories that are not required to be locally cartesian closed. So this should capture the examples you're interested in.

Nathanael Arkor (Apr 04 2025 at 09:11):

@David Corfield: thanks for the references! I was aware there was a connection between polynomials and Tambara functors, but had not yet looked into it closely; I shall have to do so. Street's recent paper is indeed relevant to our work, though I haven't yet digested it entirely. Street's perspective of polynomials as iterated spans seems closed related to our perspective of polynomials as iterated bridges (though by working in the setting of double categories, bridges allow us to give a more fine-grained iterative construction). We will certainly clarify the precise connection in the final paper.

David Corfield (Apr 04 2025 at 10:56):

One more related reference, according to Joachim Kock Notes on polynomial functors, p. 192, Tambara first looked at a category which is the Lawvere theory for commutative semirings, agreeing with Strickland:

One way to think about the definition of Mackey functors (for a finite group G) is as follows. Take the Lawvere theory A for (commutative) semigroups, categorify it, make it G-equivariant, decategorify, and then take the category of models for the resulting (multisorted) theory. This category is just the category of semigroup-valued Mackey functors, so it contains the more usual category of group-valued Mackey functors.

...

The category of Tambara functors can be defined along the same lines as suggested above for Mackey functors. One simply starts with the Lawvere theory U for semirings instead of the theory A for semigroups.

David Corfield (Apr 04 2025 at 14:09):

That passage by Kock doesn't seem to be there now.

David Corfield (Apr 04 2025 at 14:19):

The term 'bispan' seems to be current in the higher-algebraic world, as with

• Elden Elmanto, Rune Haugseng, On distributivity in higher algebra I: The universal property of bispans

John Baez (Apr 04 2025 at 15:44):

Could anyone please explain what "Mackey functors" are, in a way that a non-category theorist interested in finite groups and their representations and actions would understand? I am not wholly lacking category theory, but Street's explanation of Mackey functors, even in the presumably original example of finite groups, eluded me. I also don't understand Joachim Kock's explanation:

One way to think about the definition of Mackey functors (for a finite group G) is as follows. Take the Lawvere theory A for (commutative) semigroups, categorify it, make it G-equivariant, decategorify, and then take the category of models for the resulting (multisorted) theory.

I can follow each word, but I don't understand why Mackey would have wanted to do something like this. I've met Mackey and read some of his books, and he always struck me as a very down-to-earth representation theorist. So Kock's explanation must be a distillation of something a representation theorist might want to look at.

David Corfield (Apr 04 2025 at 15:47):

How about from the article I just mentioned?:

image.png

John Baez (Apr 04 2025 at 16:09):

Thanks, that's quite helpful. I guess I need to read some traditional writing on Mackey functors to see how they're used, and why they boil down to finite-product-preserving functors from the bicategory of spans of G-sets to Set (or, as the footnote in that quote says, more traditionally AbGp).

This paper gives a pile of examples of Mackey functors, which makes it clear to me what they're like and why they are interesting:

• Peter Webb, A guide to Mackey functors.

The axiomatics here are clunky (though intuitive if one is familiar with the examples they encapsulate), so it makes a lot of sense that "finite-product-preserving functors from the bicategory of spans of G-sets to AbGp" is a better way of saying all of this stuff.

Nathanael Arkor (Apr 04 2025 at 16:29):

David Corfield said:

The term 'bispan' seems to be current in the higher-algebraic world, as with

• Elden Elmanto, Rune Haugseng, On distributivity in higher algebra I: The universal property of bispans

Thanks, I wasn't aware of this work!

David Michael Roberts (Apr 05 2025 at 05:57):

@Nathanael Arkor I was more wondering what is the precise relation when you do the construction you mentioned in your talk to a non-LCC category, to get all polynomials (not just the ones involving powerful middle maps), because then you still have some polynomial functors already, as well as the ones coming from the freely-generated thing. There should be a relation between the existing polynomials and the freely adjoined ones, no?

Nathanael Arkor (Apr 06 2025 at 14:23):

• Given a LCC category $\mathcal E$, one can form a double category $\mathbb P\mathbf{oly}(\mathcal E)$ whose loose morphisms are polynomials (whose middle map is necessarily exponentiable) and we have a functor $\mathbb P\mathbf{oly}(\mathcal E) \to \mathbb S\mathbf{lice}(\mathcal E)$ sending each polynomial to its associated polynomial functor.
• When $\mathcal E$ has pullbacks, but is not necessarily LCC, one can form a double category $\mathbb P\mathbf{oly}_{\text{exp}}(\mathcal E)$ whose loose morphisms are polynomials with middle map exponentiable and we have a functor $\mathbb P\mathbf{oly}_{\text{exp}}(\mathcal E) \to \mathbb S\mathbf{lice}(\mathcal E)$ sending each polynomial to its associated polynomial functor.
• When $\mathcal E$ has pullbacks, but is not necessarily LCC, one can also form a covirtual double category $\mathbb P\mathbf{oly}(\mathcal E)$ whose loose morphisms are polynomials (whose middle map is not necessarily exponentiable). In this case, there is only a partial functor $\mathbb P\mathbf{oly}(\mathcal E) \rightharpoonup \mathbb S\mathbf{lice}(\mathcal E)$, defined on those loose morphisms in which the middle map is exponentiable. When $\mathcal E$ is LCC, this covirtual double category is actually a double category (the one in the first bullet point). Furthermore, it contains $\mathbb P\mathbf{oly}_{\text{exp}}(\mathcal E)$ as a full sub-double category.

Nathanael Arkor (Apr 06 2025 at 14:26):

It's important to distinguish between the polynomials and the polynomial functors: the $\mathbb P\mathbf{oly}$ construction is not adjoining new polynomial functors. In fact, the structure that it freely adjoins to $\mathcal E$ is "absolute" in an appropriate sense. If you freely added that structure (companions, conjoints, and correlates) again, it would not change the double category (up to equivalence).

Nathanael Arkor (Apr 06 2025 at 14:28):

If you just care about polynomial functors in a non-LCC $\mathcal E$, then there is not a significant advantage to working with $\mathbb P\mathbf{oly}(\mathcal E)$ over $\mathbb P\mathbf{oly}_{\text{exp}}(\mathcal E)$. However, there may be other constructions one cares about doing with polynomials for which exponentiability is unnecessary, in which case $\mathbb P\mathbf{oly}(\mathcal E)$ is more convenient.

Nathanael Arkor (Apr 06 2025 at 14:31):

Perhaps the point of confusion is that it sounded like we get a total functor $\mathbb P\mathbf{oly}(\mathcal E) \to \mathbb S\mathbf{lice}(\mathcal E)$ for non-LCC $\mathcal E$, and so there are some polynomial functors in the image of the functor for those polynomials whose middle map is not exponentiable? This isn't the case, because when $\mathcal E$ is not LCC, the double category $\mathbb S\mathbf{lice}(\mathcal E)$ does not have all correlates, and so we cannot invoke the universal property.

Nathanael Arkor (Apr 06 2025 at 14:31):

Let me know if I didn't answer your question, though. I may have misunderstood what you were asking.

David Corfield (Apr 06 2025 at 17:16):

Nathanael Arkor said:

The term 'bispan' seems to be current in the higher-algebraic world, as with

• Elden Elmanto, Rune Haugseng, On distributivity in higher algebra I: The universal property of bispans

Thanks, I wasn't aware of this work!

If double categories prove to be a good way to treat polynomials, Rune Haugseng should find your ideas very interesting since he's also the one who introduced [[double infinity-categories]].

David Michael Roberts (Apr 07 2025 at 06:36):

@Nathanael Arkor I think the point you make about the partial functor Poly(E) -> Slice(E), for Poly(E) the covirtual double category is what I was after. I was thinking that this was in fact an everywhere-defined functor, and was curious to know what difference there is in the resulting polynomial functors if you go via Poly_exp(E) vs via Poly(E).