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Stream: event: MIT Categories Seminar

Topic: May 28: David Spivak's talk

Paolo Perrone (May 26 2020 at 17:42):

Hello all, here's the stream for David's talk, "Polynomial functors II: Seven wonders of the composition product".

Paolo Perrone (May 26 2020 at 17:42):

Youtube live stream:
https://youtu.be/bLVHO5LxUY0

Paolo Perrone (May 26 2020 at 17:42):

Zoom meeting:
https://mit.zoom.us/j/280120646
Meeting ID: 280 120 646

Paolo Perrone (May 27 2020 at 19:49):

The slides are available, here: http://math.mit.edu/~dspivak/informatics/talks/poly2_wondersMIT2020.pdf

Jacques Carette (May 28 2020 at 13:20):

A lot of the wonderful properties of Poly seems to be 'shared' with Species (and also of other categories from concurrency theory based on [Inj, Set] ). I've been slowly exploring what makes all those monoidal structures on [A, B] work, as a way to understand how they all fit together.

Gershom (May 28 2020 at 15:28):

@Jacques Carette can you name some of the other categories from concurrency theory of which you are thinking? :-)

Jacques Carette (May 28 2020 at 15:29):

I'm mainly thinking of the functor category I mentioned, from (finite sets + injections) to Set. It seems to be quite pervasive.

Gershom (May 28 2020 at 15:36):

Is this the same thing as the schanuel topos (https://ncatlab.org/nlab/show/Schanuel+topos)? I know of a variety of uses in computer science, but I'm sure which concurrency theory applications you're thinking of?

Nathanael Arkor (May 28 2020 at 15:38):

The Schanuel topos is a subcategory of [Inj, Set], where the functors are pullback-preserving.

Jacques Carette (May 28 2020 at 15:43):

I'll dig up some references - I can't immediately find good ones.

Paolo Perrone (May 28 2020 at 15:50):

Hello all! We start in 10 minutes.

Paolo Perrone (May 28 2020 at 15:59):

30 seconds!

Cole Comfort (May 28 2020 at 15:59):

Do you have a link?

Cole Comfort (May 28 2020 at 16:00):

Never mind!

Nikolaj Kuntner (May 28 2020 at 16:15):

What does Poly not have that Set has?

Brian Pinsky (May 28 2020 at 16:20):

Can you go over what morphisms in poly are again?

Brian Pinsky (May 28 2020 at 16:21):

(in my head it's a generalization of "divides" for classic polynomials but I'm not sure that's right)

Brian Pinsky (May 28 2020 at 16:22):

Okay thanks

Jacques Carette (May 28 2020 at 16:58):

I first encountered Inj in A fully abstract model for the Pi-calculus.

Gershom (May 28 2020 at 17:00):

The discussion of symmetry and lists reminds me of this old post on combinatorics from brent which is about ordered vs. unordered gadgets. List is the sum over ordered n-tuples, so one also may want to consider sets as the sum over unordered n-tuples: https://byorgey.wordpress.com/2012/08/24/unordered-tuples-and-type-algebra/

Jacques Carette (May 28 2020 at 17:04):

https://www.cis.upenn.edu/~sweirich/papers/yorgey-thesis.pdf

Nathanael Arkor (May 28 2020 at 17:12):

Is there an expectation that all of this structure should also arise in the context of general polynomial functors on locally cartesian-closed categories, or did anything described in the talk particularly rely on Set?

Mike Shulman (May 28 2020 at 17:20):

I missed the talk but the slides are nice, thanks! The definition of cofunctor is reminiscent of cartesian liftings in a fibration. Can a cofunctor $C \to D$ be identified with something like a span $D \leftarrow E \to C$ of functors in which the first leg is a discrete opfibration and the second leg is bijective on objects?

Nathanael Arkor (May 28 2020 at 17:26):

@Mike Shulman: a similar characterisation of (internal) cofunctors is given in Clarke's Internal Lenses as Functors and Cofunctors.

Mike Shulman (May 28 2020 at 17:31):

Not just similar but identical! Thanks.

Nathanael Arkor (May 28 2020 at 17:37):

The best kind of similar :big_smile:

Joachim Kock (May 28 2020 at 19:22):

Mike Shulman said:

Can a cofunctor $C \to D$ be identified with something like a span $D \leftarrow E \to C$ of functors in which the first leg is a discrete opfibration and the second leg is bijective on objects?

That's right. One interesting thing about this is that this kind of span is precisely those that induce coalgebra homomorphisms on the incidence coalgebras. It works the same for decomposition spaces more generally than category objects, but then the bijective-on-objects should be
bijective on objects and inner Kan fibration. (For category objects the inner-Kan-fibration condition is automatic.)

Joachim Kock (May 28 2020 at 19:23):

Nathanael Arkor said:

Is there an expectation that all of this structure should also arise in the context of general polynomial functors on locally cartesian-closed categories, or did anything described in the talk particularly rely on Set?

Yes, it should be exactly the same in the general case.

Joachim Kock (May 28 2020 at 19:31):

Nathanael Arkor said:

A similar characterisation of (internal) cofunctors is given in Clarke's Internal Lenses as Functors and Cofunctors.

Thanks, this is very nice.

Joachim Kock (May 28 2020 at 19:35):

Joachim Kock said:

this kind of span is precisely those that induce coalgebra homomorphisms on the incidence coalgebras

Sorry, I was too fast: not exactly the same, but an important example of. To get coalgebra homomorphisms, it is enough for the forward leg to be culf. Discrete opfibration is an example of that.

Todd Trimble (May 28 2020 at 19:41):

What does culf mean?

Joachim Kock (May 28 2020 at 19:51):

Todd Trimble said:

What does culf mean?

Sorry --- it's for conservative and unique-lifting-of-factorisations, also known as discrete Conduché fibration. (In higher settings, the discrete-Conduché-fibration terminology becomes a bit akward, because discreteness is no longer appropriate (and in any case these things were discovered before Conduché.) Actually 'conservative' is implied by 'ulf' for categories, but in more general simplicial situations, it is nice to have it as a separate condition.

Paolo Perrone (May 28 2020 at 22:13):

Hey all, here's the permanent video:
https://youtu.be/3AOGDTr1zrY

Bryce Clarke (May 28 2020 at 22:20):

Mike Shulman said:

The definition of cofunctor is reminiscent of cartesian liftings in a fibration.

The link between cofunctors and opcartesian lifts for split opfibrations is the topic of my recent preprint: Internal split opfibrations and cofunctors

Jules Hedges (May 29 2020 at 09:35):

I didn't notice before, this is the Topos Institute youtube channel, a sign of the future

Joachim Kock (May 30 2020 at 21:32):

CORRECTION

In my comment after David's talk, I said there is a fifth monoidal structure, namely the Hadamard product (which for power series is $(\sum a_i z^i)(\sum b_i z^i) = \sum (a_i b_i) z^i$). Unfortunately it's not true for the whole category $\mathbf{Poly}$ David talks about. The Hadamard product is functorial only in cartesian natural transformations, not in general ones :-(

For example: we have $y^1 @ y^1 = y^1$; there is a map $y^1 \to y^3$ (induced by $3\to 1$); and we have $y^1 @ y^3 = 0$. But there is no map from $y^1$ to $0$.

Sorry about the mistake. (I am too used to working only with cartesian natural transformations, which are the ones relevant for operad theory).

Nathanael Arkor (May 30 2020 at 21:46):

That actually resolves a difficulty I was having yesterday defining the Hadamard product; I thought I must be missing something because I couldn't get it to work. Thank you for clarifying!

Morgan Rogers (he/him) (May 31 2020 at 11:04):

@Jacques Carette did you manage to dig up the further monoidal product you mentioned in the questions at the end of the talk?
EDIT: Ah, I see, that's what the thesis link was for.

Jacques Carette (May 31 2020 at 11:53):

Right, that link was thrown there live, without explanation. The monoidal structure is called the 'arithmetic product' by those who first pointed it out (Maia and Mendez - https://www.sciencedirect.com/science/article/pii/S0012365X07007960) from 2008. The link to Brent's thesis is because he explains it nicely. @Joachim Kock said that this might also be the Dirichlet product?

I've also found the set of slides https://www.slideshare.net/vieplivee/mit-0309-nopause that introduces the relation with a product on graphs to be quite interesting.

Nathanael Arkor (May 31 2020 at 11:55):

The arithmetic product of species is also called the Dirichlet product, yes.

Morgan Rogers (he/him) (May 31 2020 at 15:13):

Did anyone work out if that restricts to Poly in a nice way? It certainly doesn't in the finitary world, but we have that much more flexibility here.

Nathanael Arkor (May 31 2020 at 15:19):

David described the Dirichlet product of polynomials in his talk (the tensor product $\otimes$).

Morgan Rogers (he/him) (May 31 2020 at 15:30):

Oh my mistake, I was looking at the Dirichlet convolution, which from its unit I gather is a distinct concept, and isn't even related to polynomials...

John Baez (May 31 2020 at 18:20):

By the way, I listed the 5 main monoidal structures on species on the nLab.

Joshua Meyers (May 31 2020 at 21:24):

Does anyone know why Poly is the subcategory of [Set,Set] spanned by functors that preserve connected limits?

Nathanael Arkor (May 31 2020 at 21:43):

Polynomial functors and polynomial monads gives proofs/references for the equivalence of this characterisation with others in Section 1.18.

Todd Trimble (May 31 2020 at 21:44):

If a functor preserves connected limits and the terminal, then it preserves all limits. So if $F: Set \to Set$ preserves connected limits, then the evident induced functor $F^\ast: Set \to Set/F(1)$, taking $X$ to $F(!): F(X) \to F(1)$, preserves all limits. Then, pulling back along each element $x: 1 \to F(1)$ gives a functor $x^\ast F^\ast: Set \to Set/(F(1) \to Set/1 \simeq Set$ which preserves limits, hence is representable, say a functor $\hom(S_{F, x}, -)$. In the end we get $F \cong \sum_{x \in F(1)} \hom(S_{F, x}, -)$.

Joshua Meyers (Jun 01 2020 at 05:22):

Thanks @Todd Trimble. Why does the pullback $x^{*}$ preserve limits? Why must a set functor preserving limits be representable?

Joachim Kock (Jun 01 2020 at 07:02):

Nathanael Arkor asked:

Is there an expectation that all of this structure should also arise in the context of general polynomial functors on locally cartesian-closed categories, or did anything described in the talk particularly rely on Set?

and I replied

Yes, it should be exactly the same in the general case.

It was silly of me to answer like this.

There are several subtleties or even problems, when passing from Set to more general locally cartesian categories: for example sometimes you have to demand that some objects are discrete/complemented/decidable; you have to make more careful distinction between internal sums and external sums; you generally have to assume natural transformations are strong (that's automatic in Set). Altogether I think my answer was bad.

Joachim Kock (Jun 01 2020 at 07:03):

I have already made several comments that were too fast :-( I should take a break.

Paolo Capriotti (Jun 01 2020 at 09:04):

Joshua Meyers said:

Thanks Todd Trimble. Why does the pullback $x^{*}$ preserve limits? Why must a set functor preserving limits be representable?

The functor $x^*$ preserves limits because it has a left adjoint, namely the functor $x_!\colon \mathsf{Set} \to \mathsf{Set}/X$ which sends a set $A$ to the map $A \to 1 \xrightarrow{x} X$.

If a functor $F\colon \mathsf{Set} \to \mathsf{Set}$ preserves limits, then it has a left adjoint $L$ by the special adjoint functor theorem, hence $F(X) \cong \mathsf{Set}(1, F(X)) \cong \mathsf{Set}(L(1), X)$, which says that $F$ is representable.