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Stream: community: our work

Topic: Paolo Perrone

Paolo Perrone (Nov 19 2022 at 14:34):

Hi all,
I've recently given this talk, "When measurable spaces don't have enough points". It was not recorded, but some people still wanted to see it, so here's an unofficial recording!

https://youtu.be/V4WzTgjbP3c

Ivan Di Liberti (Nov 19 2022 at 17:34):

Davvero bel talk.

Paolo Perrone (Nov 19 2022 at 18:02):

Grazie!

Paolo Perrone (Jul 02 2024 at 11:10):

Hi all!
Here is an extended recording of my talk at CT.
https://youtu.be/awr4RBrhh1g
It is an introductory video to the work that Ruben Van Belle and I did on martingales. Any feedback is welcome, I hope you find it interesting!

Paolo Perrone (Jul 02 2024 at 11:12):

Starting at time 36.54 there's also an introduction to the idea of conditional expectation aimed at category theorists, which some people seemed to be interested in.

Rémy Tuyéras (Jul 02 2024 at 11:40):

Thanks for sharing your work Paolo! Cool video!

Peva Blanchard (Jul 02 2024 at 21:35):

I'm really glad that I know enough category theory now to be able to follow the arguments, but I have to say thank you for all the efforts you put into making the topic accessible (drawings, explanations, mental pictures, etc.). Bravo!

And cherry on top: I love these kinds of "aha" moments realizing that "limits are limits", martingale convergence is really a categorical limit.

Kalin Krishna (Jul 03 2024 at 08:00):

Thank you for sharing. It was really an interesting talk.

David Corfield (Oct 11 2024 at 09:12):

I'm enjoying How to Represent Non-Representable Functors. This
image.png
is representing precisely this situation:
image.png
nLab: motivation for sheaves, cohomology and higher stacks.

Paolo Perrone (Oct 11 2024 at 12:57):

I'm glad you're enjoying it!
Yes, that is precisely the idea. :)

Paolo Perrone (Oct 11 2024 at 12:58):

By the way, this is not my usual field of expertise, so any feedback would be more than welcome!

Paolo Perrone (Oct 11 2024 at 13:01):

Especially, I'd like to know if I've missed some references. This idea of "virtual arrows" seems somewhat folklore, but I couldn't find it written anywhere besides that nLab page and a few related ones. (And I couldn't find any diagram chasing done this way.)
Since I couldn't find material on it, I wrote some myself. But if I've overlooked something, or if I've made any mistakes, please let me know.

Kevin Carlson (Oct 11 2024 at 18:09):

Yes, this is really excellent! As you mention, the basic principle here has been "known" as heteromorphisms for ages, but I really think this exposition is one of the first to make good use of heteromorphisms by emphasizing set functors. I particularly think this presentation has got to make weighted limits a meaningful step more approachable. I do wonder whether you have any thoughts about whether these pictures are going to be useful in enriched categories, since so much of the weighted colimit story really comes into its own in that context.

Paolo Perrone (Oct 11 2024 at 18:28):

I imagine that this could still be useful for those enrichments which are still enough like sets - like abelian groups or posets. Other types of enrichments, like real numbers (giving Lawvere metric spaces) may be too different.

Nathanael Arkor (Oct 11 2024 at 18:36):

@fosco and I taught an introductory course on category theory at the start of this year in which we took a very similar approach to representability (even using the terms "virtual object" and "virtual morphism" for the same purpose). I think one of the disadvantages in teaching the material that way was that it was quite different from other material, so there was a separation between what we were teaching and what it was convenient for the students to find in other sources. So I'm really glad to see a thorough expository reference from this perspective – I think it's a very helpful way to reason about representability (particularly when learning category theory), and I think the intuitive terminology is much better than the classical terminology.

Paolo Perrone (Oct 11 2024 at 18:48):

Nathanael Arkor said:

fosco and I taught an introductory course on category theory at the start of this year in which we took a very similar approach to representability (even using the terms "virtual object" and "virtual morphism" for the same purpose). I think one of the disadvantages in teaching the material that way was that it was quite different from other material, so there was a separation between what we were teaching and what it was convenient for the students to find in other sources. So I'm really glad to see a thorough expository reference from this perspective – I think it's a very helpful way to reason about representability (particularly when learning category theory), and I think the intuitive terminology is much better than the classical terminology.

Oh, really nice! (And funny that we both came up with the name "virtual". I was going for "imaginary" but then it sounded too weird.)
Do you have any material from those lectures? I'd love to hear what you taught.

Nathanael Arkor (Oct 11 2024 at 20:08):

I will have a look for the notes for the lectures when I'm back in my office next week :)

Peva Blanchard (Oct 12 2024 at 11:49):

I find the "virtual object" picture very nice to build intuition about presheaves. It also resonates with something that clicked for me recently about profunctors. Namely that profunctors is "just" a way to specify "morphisms" between objects from two different categories.

Tom Leinster (Oct 12 2024 at 14:20):

An underused piece of notation (in my opinion) is $m: a \not\to b$ to mean $m \in M(a, b)$, where $M: \mathbf{A}^{\mathrm{op}} \times \mathbf{B} \to \mathbf{Set}$ is a functor. (Here $\not\to$ should be a profunctory crossed arrow, which I don't know how to typeset here.) What's good about this notation is that you can compose like this:

$a' \stackrel{f}{\to} a \stackrel{m}{\not\to} b \qquad\mapsto\qquad a' \stackrel{mf}{\not\to} b$

$a \stackrel{m}{\not\to} b \stackrel{g}{\to} b' \qquad\mapsto\qquad a \stackrel{gm}{\not\to} b'$

Everything works particularly nicely if you write $M: \mathbf{A}^{\mathrm{op}} \times \mathbf{B} \to \mathbf{Set}$ as $M: \mathbf{A} \not\to \mathbf{B}$, as then the elements of $M: \mathbf{A} \not\to \mathbf{B}$ look like $m: a \not\to b$. This is the less popular of the two conventions for (bi)modules/profunctors/distributors, but it has this advantage.

Drawing elements of presheaves as arrows is a special case of this notation.

I used this notation for bimodule elements in my work on self-similarity, e.g. here, but I wouldn't be remotely surprised if others had done the same before me.

Paolo Perrone (Oct 12 2024 at 14:29):

Oh very nice, I like that.

Paolo Perrone (Oct 12 2024 at 14:44):

Reading the paper you linked, I do see some diagram chasing done with "virtual arrows". Very nice. (Of course I'll cite you in the next round of updates.)

Mike Shulman (Oct 12 2024 at 16:13):

You can paste the unicode character 0x21F8 RIGHTWARDS ARROW WITH VERTICAL STROKE to get $M:A⇸B$.

Todd Trimble (Oct 12 2024 at 19:01):

Someone (I don't remember who) explained once that the stroked arrow can be imagined as an arrow that breaks out of the barrier (symbolized by the stroke) of one category and into another.

John Baez (Oct 12 2024 at 19:17):

I tend to use ⇸ for loose arrows in a double category - like relations, spans and profunctors. I imagine the bar as negation, like as "you think this is a tight arrow? No, it's a loose arrow?"

Then the use of f:a⇸b to denote an element of a profunctor F:A⇸B would be a nice case of the microcosm principle: "to make sense a morphism from an object of one category to an object of another, we need a profunctor from one category to another". Jim Dolan might say f rides the profunctor F, like a person rides a horse, or a map between monoid objects in two different monoidal categories rides a lax monoidal functor between these monoidal categories.

Paolo Perrone (Oct 12 2024 at 21:24):

These are all such nice ideas!
Is there any more material, no matter how informal, where they are pursued?

Mike Shulman (Oct 12 2024 at 22:02):

And an element of a profunctor is also, itself, a loose morphism in a double category, namely the double category of pointed categories and pointed profunctors.

John Baez (Oct 12 2024 at 22:20):

Paolo Perrone said:

These are all such nice ideas!
Is there any more material, no matter how informal, where they are pursued?

I think a lot of it is "folklore" passed down through conversations.

Mike Shulman (Oct 12 2024 at 23:27):

There are some references at [[heteromorphism]].

John Baez (Oct 13 2024 at 00:02):

David Ellerman's papers on heteromorphisms are a strong attempt to push for that way of thinking.

Paolo Perrone (Oct 13 2024 at 09:22):

Oh wow, I wasn't aware of Ellerman's writings. That's a gold mine!

Morgan Rogers (he/him) (Oct 13 2024 at 20:14):

@Ivan Di Liberti and I used diagrams for an "evaluation" profunctor in our last paper. We used a diagram with a dotted line down the middle, where morphisms in the respective categories were on the respective sides and heteromorphisms crossed from one side to the other. We couldn't find examples of these diagrams around, although I was sure I had seen such things illustrating the ideo of heteromorphism beforehand.

Paolo Perrone (Oct 13 2024 at 20:25):

Thank you!

Matteo Capucci (he/him) (Oct 22 2024 at 09:43):

I might add myself to the list of people who have been using this idea for a while, although never in (published) writing. Specifically you've seen me doing that if we ever talked about categorical systems theory, where the main structure of interest are modules over double categories, which are the double version of modules over categories, ie (co)presheaves.
Thus given such a module $\bf Sys$ over the double category $\mathbb{C}$, I usually draw elements of ${\bf Sys}(A)$ as bubbles with a dangling wire named $A$, to which (loose) arrows of $\mathbb{C}$ can be attached (and I draw them as boxes). 'Functoriality' is then the rule that a bubble $S \in {\bf Sys}(A)$ can absorb any arrow $p:A \nrightarrow B$ to become a bubble $S \cdot p \in {\bf Sys}(B)$.

Matteo Capucci (he/him) (Oct 22 2024 at 09:47):

I found some diagrams in a draft:
image.png
Functoriality:
image.png
It also works for monoidal structure:
image.png

Matteo Capucci (he/him) (Oct 22 2024 at 09:48):

Notice that, ignoring the double categorical features, these string diagrams for $F:C \to \bf Set$ are roughly string diagrams in $\int F$

Paolo Perrone (Oct 22 2024 at 13:39):

Very nice stuff!

Paolo Perrone (Oct 22 2024 at 13:39):

As it's hopefully clear from the paper, I'm not claiming I've invented any of this.
I'm just writing it down in detail and showing how to use it, so that people in the future (learners, practitioners, confused or curious readers of our papers...) have somewhere to start.

Paolo Perrone (Oct 22 2024 at 13:39):

By the way, if anyone is looking for references, the first published work where I could find ideas like these is Section 2.2 of Bodo Pareigis, Categories and Functors, 1970.
I also recommend David Ellerman's papers (such as this one) where one can find an in-depth discussion of the idea and its history. Ellerman also seems to be the person who coined the name heteromorphism.

Matteo Capucci (he/him) (Oct 22 2024 at 14:17):

Paolo Perrone said:

As it's hopefully clear from the paper, I'm not claiming I've invented any of this.
I'm just writing it down in detail and showing how to use it, so that people in the future (learners, practitioners, confused or curious readers of our papers...) have somewhere to start.

Of course! You did a great job at that