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Stream: event: ACT@UCR

Topic: May 27th: Simon Willerton

John Baez (May 26 2020 at 03:48):

In the ninth talk of the ACT@UCR seminar, Simon Willerton will tell us about a categorical approach to the Legendre transform, and its connection to tropical algebra.

He will give his talk on Wednesday May 27th at 5 pm UTC, which is 10 am in California, or 1 pm on the east coast of the United States, or 6 pm in the UK. It will be held online via Zoom, here:

https://ucr.zoom.us/j/607160601

You can see his slides here:

• Simon Willerton, The Legendre–Fenchel transform from a category theoretic perspective.

Abstract. The Legendre-Fenchel transform is a classical piece of mathematics with many applications. In this talk I’ll show how it arises in the context of category theory using categories enriched over the extended real numbers $\overline{ \mathbb{R}}:=[-\infty,+\infty]$. It turns out that it arises out of nothing more than the pairing between a vector space and its dual in the same way that the many classical dualities (e.g. in Galois theory or algebraic geometry) arise from a relation between sets.

I won’t assume knowledge of the Legendre-Fenchel transform.

John Baez (May 26 2020 at 16:21):

Simon's talk is based on this paper:

• Simon Willerton, The Legendre-Fenchel transform from a category theoretic perspective.

Also see his blog article:

• Simon Willerton, The nucleus of a profunctor: some categorified linear algebra, The n-Category Café.

Jules Hedges (May 27 2020 at 10:57):

Unless I'm confused, this talk (and all the future ones) aren't on the categorytheory.world calendar

Joe Moeller (May 27 2020 at 16:22):

who's in charge of putting things on that calendar?

sarahzrf (May 27 2020 at 18:02):

wtf i love analysis now

sarahzrf (May 27 2020 at 18:21):

so the thing i was saying about groupoid cores—

sarahzrf (May 27 2020 at 18:22):

ive had this definition in my depleted category theory development

Definition core_rel {X} (R : X -> X -> Prop) : X -> X -> Prop := fun a1 a2 => R a1 a2 /\ R a2 a1.

sarahzrf (May 27 2020 at 18:22):

basically i've figured that if you have a preorder $R$, then its core is given by $a \mathrel C b \iff a \mathrel R b \land b \mathrel R a$

sarahzrf (May 27 2020 at 18:23):

which seems rather related to $d'(x, y) = \max \{d(x, y),\ d(y, x)\}$

eric brunner (May 27 2020 at 18:26):

q1: pointer to "concept lattice"

eric brunner (May 27 2020 at 18:27):

q2: pointer to paper(s) on server placement(s).

John Baez (May 27 2020 at 18:30):

Joe Moeller said:

who's in charge of putting things on that calendar?

Maybe @Daniel Geisler? He got the initial list of talks we gave him, but probably not the last few talks of this series, since we hadn't nailed them down.

John Baez (May 27 2020 at 18:31):

Hi, Simon!

John Baez (May 27 2020 at 18:32):

I'd like to get more intuition in the case of ordinary categories C, what happens when you take the nucleus of the profunctor hom: ${C}^{\mathrm{op}} \times C \rightarrow$ Set.

Simon Willerton (May 27 2020 at 18:32):

Hi!

Simon Willerton (May 27 2020 at 18:32):

t.eric.brunner said:

q2: pointer to paper(s) on server placement(s).

M. Chrobak and L. L. Larmore, Generosity helps or an 11-competitive algorithm for three servers, Journal of Algorithms 16 (1994), 234–263.

John Baez (May 27 2020 at 18:33):

Wow, Marek Chrobak is in computer science at UCR!

sarahzrf (May 27 2020 at 18:33):

i think maybe what i'm describing up there has something to do with taking the category $C^{\mathrm{op}} \times C$

sarahzrf (May 27 2020 at 18:33):

oh, how is ucr's cs dept? :eyes:

sarahzrf (May 27 2020 at 18:33):

i should start making a list of schools to look into...

eric brunner (May 27 2020 at 18:34):

thanks!

John Baez (May 27 2020 at 18:36):

sarahzrf said:

oh, how is ucr's cs dept? :eyes:

I have no idea. The only person I talk to there is Tom Payne, who used to be in logic in the math department, and was then head of CS for a while, and is now retired I believe. I don't think there's anyone doing category theory there.

John Baez (May 27 2020 at 18:39):

sarahzrf said:

basically i've figured that if you have a preorder $R$, then its core is given by $a \mathrel C b \iff a \mathrel R b \land b \mathrel R a$

I believe we can define a dagger-category in the enriched setting - an enriched category $C$ with an enriched contravariant functor $\dagger: C \to C$ that's the identity on objects and squares to the identity. And then I believe a $[0,\infty]$-enriched dagger category is a Lawvere metric space where the metric is symmetric.

John Baez (May 27 2020 at 18:40):

I bet enriched dagger-categories are more general than "enriched groupoids" in general (does anyone talk about enriched groupoids?).

eric brunner (May 27 2020 at 18:40):

thanks! now i see their literature.

sarahzrf (May 27 2020 at 18:40):

yeah i seem to recall seeming reference to the same point on the nlab

sarahzrf (May 27 2020 at 18:40):

i think even on the page for lawvere metric spaces... :)

John Baez (May 27 2020 at 18:41):

I bet a Truth-enriched dagger-category is a set with an equivalence relation. Hmm, but this may also be Truth-enriched groupoid.

John Baez (May 27 2020 at 18:41):

So in that case the notions of enriched dagger-category and enriched groupoid may agree.

sarahzrf (May 27 2020 at 18:41):

a "setoid"

John Baez (May 27 2020 at 18:41):

I'm still hoping for you to answer my question, @Simon Willerton:

John Baez (May 27 2020 at 18:42):

John Baez said:

I'd like to get more intuition in the case of ordinary categories C, what happens when you take the nucleus of the profunctor hom: ${C}^{\mathrm{op}} \times C \rightarrow$ Set.

Simon Willerton (May 27 2020 at 18:42):

I trying to think and remember.

John Baez (May 27 2020 at 18:42):

Okay!

Simon Willerton (May 27 2020 at 18:42):

It's a while since I thought of non-poset-enriched examples!

Gershom (May 27 2020 at 18:43):

My sense from the discussion during the talk was this is the Isbell envelope?

John Baez (May 27 2020 at 18:44):

For a second I was getting excited because Dusko was raising the subject of some way to simultaneously complete and cocomplete a category, that was more "even-handed" than taking presheaves.

John Baez (May 27 2020 at 18:44):

Maybe it is the Isbell envelope; I don't know anything about that or what it's like.

Paolo Perrone (May 27 2020 at 18:44):

John Baez said:

I bet enriched dagger-categories are more general than "enriched groupoids" in general (does anyone talk about enriched groupoids?).

There's a little here: https://ncatlab.org/nlab/show/metric+space#LawvereMetricSpace

John Baez (May 27 2020 at 18:45):

Nice:

Note that when enriching over a cartesian monoidal poset, there is no difference between a †-category and a groupoid, so ultrametric spaces can also be regarded as enriched groupoids, which is perhaps a more familiar concept.

Gershom (May 27 2020 at 18:46):

(i guess more like the category of saturated objects in the isbell envelope, hmm...)

eric brunner (May 27 2020 at 18:47):

i see references in the ml literature, but i was expecting something more logic, and less data.

John Baez (May 27 2020 at 18:48):

I have no idea what the Isbell envelope is, or "saturated objects", or what they're like.

John Baez (May 27 2020 at 18:48):

I guess I need to read about this stuff.

eric brunner (May 27 2020 at 18:49):

thanks buckets to simon for an interesting hour and then some.

John Baez (May 27 2020 at 18:50):

I'll ask another question: how can we get this approach to the Legendre transform to generalize to the Laplace transform? It's already known that the Laplace transform has the Legendre transform as a "zero-temperature limit".

There's a rig depending on temperature $T \in [0,\infty]$ s.t. for $T > 0$ we can use this rig to define a Laplace transform, and as $T \to 0$ it converges to the Legendre transform. This is the idea of "idempotent analysis".

Gershom (May 27 2020 at 18:50):

@sarahzrf in your depleted stuff, do you have a nice characterization in simple order-theoretic terms of what a profunctor is? I have a decent sense, I think, but if you worked out the details it would be interesting.

sarahzrf (May 27 2020 at 18:51):

yeah it's a relation satisfying a "rule of consequence"

$\frac{a' \le a \qquad a \mathrel R b \qquad b \le b'}{a' \mathrel R b'}$

John Baez (May 27 2020 at 18:52):

You don't need the "latex" here, Simon. Instead you need double dollar signs!

John Baez (May 27 2020 at 18:52):

:smiling_devil:

Simon Willerton (May 27 2020 at 18:53):

Yeah, just looked at the source of a post.

John Baez (May 27 2020 at 18:53):

Truly irritating.

John Baez (May 27 2020 at 18:53):

We do this just to spot outsiders.

John Baez (May 27 2020 at 18:53):

I guess it's called a "shibboleth".

Simon Willerton (May 27 2020 at 18:55):

I was trying to say that in the general case you have an adjunction $L:[C^{op}, V] \leftrightarrows [C,V]^{op}:R$.

Simon Willerton (May 27 2020 at 18:56):

If $V$ is a poset then the monad and comonad are both idempotent which makes things much easier.

John Baez (May 27 2020 at 18:58):

Oh, okay.

Simon Willerton (May 27 2020 at 18:58):

In the non-poset case such as for Set-categories(!) you don't really want to look at the fixed points of the adjunction, but rather weak fixed points. And I think that's what the Isbell envelope is.

Simon Willerton (May 27 2020 at 18:59):

I did think a lot about this in the past with my student Jonathan Elliott.

John Baez (May 27 2020 at 18:59):

Nice! By "weak fixed point" do you mean objects equipped with an isomorphism...?

Gershom (May 27 2020 at 18:59):

The isbell envelope page on n-lab suggests to consider the simpler case where C is concrete over Set.

John Baez (May 27 2020 at 19:01):

I'll check it out. Simon's suggestion seems nice because $[C^{\mathrm{op}},Set]$ is presheaves, which is the free cocompletion, but then we're tweaking that a bit... in a way I don't really grok.

Simon Willerton (May 27 2020 at 19:02):

Either weak or possibly lax. You wouldn't take, say, a presheaf $f\in [C^{op},Set]$ such that $RL(f)=f$ but rather take $f\in [C^{op},Set]$ together with an isomorphism or possibly just a morphism $f\to RL(f)$

Simon Willerton (May 27 2020 at 19:02):

(Or the arrow going the other way.)

Simon Willerton (May 27 2020 at 19:06):

Or maybe that's not even right! You want $f\in [C^{op},Set]$, $g\in [C,Set]^{op}$ and a natural transforamtion $f(-) \times g(-) \Rightarrow Hom(-,-)$.

John Baez (May 27 2020 at 19:06):

Is there a place I can read about this? Maybe something Jonathan Elliott wrote?

Simon Willerton (May 27 2020 at 19:07):

In his thesis, he just thought about enriching over quantales as we got nowhere enriching over general categories, and found something meaty in fuzzy truth values.

Simon Willerton (May 27 2020 at 19:08):

There's the nlab pages.

John Baez (May 27 2020 at 19:08):

Meaty and fuzzy... :cow:

John Baez (May 27 2020 at 19:09):

Btw, Simon, I'm shifting over to spending more and more time writing "expository/research" math stuff.

John Baez (May 27 2020 at 19:10):

The chance that I can put some serious work into helping you with that book on "applied enriched category theory" keeps going up.

Simon Willerton (May 27 2020 at 19:10):

John Baez said:

I'll ask another question: how can we get this approach to the Legendre transform to generalize to the Laplace transform? It's already known that the Laplace transform has the Legendre transform as a "zero-temperature limit".

There's a rig depending on temperature $T \in [0,\infty]$ s.t. for $T > 0$ we can use this rig to define a Laplace transform, and as $T \to 0$ it converges to the Legendre transform. This is the idea of "idempotent analysis".

I think this relates to the questions that were coming up with Tai-Dannae the other week involving getting a category that is modeling the real number with usual arithmetic, not max-plus.

Simon Willerton (May 27 2020 at 19:11):

John Baez said:

The chance that I can put some serious work into helping you with that book on "applied enriched category theory" keeps going up.

Bruce was telling me off yesterday for not having finished it :-)

John Baez (May 27 2020 at 19:12):

Good!

John Baez (May 27 2020 at 19:12):

I'm not gonna publicly commit myself (or you) to finishing this book here, but it seems like a good thing to write.

John Baez (May 27 2020 at 19:12):

The material is so beautiful.

Simon Willerton (May 27 2020 at 19:12):

Indeed.

Simon Willerton (May 27 2020 at 19:13):

Where there many questions coming up during the talk that didn't make it over here?

John Baez (May 27 2020 at 19:13):

Not lots. About 3 reference-type questions.

John Baez (May 27 2020 at 19:14):

I'm afraid not everyone makes it over here.

Simon Willerton (May 27 2020 at 19:16):

I often don't make it here as I am usually starting to think about dinner!

John Baez (May 27 2020 at 19:16):

Anyway, I'm finally eager to think about Isbell duality and Isbell envelopes and such.

John Baez (May 27 2020 at 19:17):

Your example of the Dedekind completion grabbed my attention.

John Baez (May 27 2020 at 19:17):

Maybe it's starting to be dinner-time (for you) now...

Simon Willerton (May 27 2020 at 19:17):

That's great.

Simon Willerton (May 27 2020 at 19:18):

I really want to think about connections with optimal transport stuff, as I've mentioned before.

Christian Williams (May 27 2020 at 19:20):

Okay, you don't need to stay; I was just trying to work out what's going on with the adjunction. This stuff is really cool, and the applications are really impressive.

Christian Williams (May 27 2020 at 19:20):

Given $R:A^{op}\otimes B\to \mathrm{V}$, we can curry in two ways: $R_1:A^{op}\to [B,\mathrm{V}]$ and $R_2:B\to [A^{op},\mathrm{V}]$. Then we also want to "op" $R_1$, so that it goes to the "opcopresheaves".

John Baez (May 27 2020 at 19:21):

That optimal transport stuff sounds cool, Simon. It reminds me slightly of that "earth-mover distance" between probability distributions.

Simon Willerton (May 27 2020 at 19:23):

One cool thing that I didn't say is that in the convex hull of a function you really are writing it as a colimit (sum?) of hyperplanes (or affine functions). The affine functions are the copresheaves (functions on V) are the ones which are representable by elements of Vdual.

sarahzrf (May 27 2020 at 19:23):

John Baez said:

I'll check it out. Simon's suggestion seems nice because $[C^{\mathrm{op}},Set]$ is presheaves, which is the free cocompletion, but then we're tweaking that a bit... in a way I don't really grok.

well, $[C, Set]^{\mathrm{op}}$ is the free completion

Christian Williams (May 27 2020 at 19:23):

Then by the properties of completion and cocompletion, we can right-extend to get $R_* = \mathrm{Ran}_{y}R_1^{op}: [A,\mathrm{V}]^{op}\to [B^{op},\mathrm{V}]$ and left-extend to get $R^*: \mathrm{Lan}_y R_2: [B^{op},\mathrm{V}]\to [A,\mathrm{V}]^{op}$.

Christian Williams (May 27 2020 at 19:23):

whoops, not quite right. it's easy to get tripped up on the op's

sarahzrf (May 27 2020 at 19:23):

i like $[C, Set]^{\mathrm{op}}$ honestly—it's actually quite a natural category to consider

Christian Williams (May 27 2020 at 19:24):

natural how?

sarahzrf (May 27 2020 at 19:24):

well

Simon Willerton (May 27 2020 at 19:24):

The Earth mover distance is exactly the thing we were discretizing when we were talking about bakeries and cafes. This is what goes into the Kantorovich duality that Gordon mentioned.

sarahzrf (May 27 2020 at 19:24):

if a presheaf is something that you can map into from objects of C, then its morphisms should be natural transformations—they tell you how to compose with a morphism from an object of C

sarahzrf (May 27 2020 at 19:24):

if a copresheaf is something that you can map out of, into an object of C...

sarahzrf (May 27 2020 at 19:25):

then the morphisms of copresheaves should go the opposite direction from natural transformations—you should be able to compose them with morphisms into objects of C!

sarahzrf (May 27 2020 at 19:25):

well, that's a long-winded way of saying that if you put an ^op on [C, Set], then your yoneda embedding into copresheaves is covariant again

Simon Willerton (May 27 2020 at 19:26):

Christian, what you're writing is looking true. Is there a question?

Christian Williams (May 27 2020 at 19:26):

yeah, that makes sense. it would be great to have a more concrete intuition. I like to think of natural transformations as inference rules.

John Baez (May 27 2020 at 19:26):

Thanks for writing it, Christian! I found the coend formula less clear than this Kan extension way of thinking.

John Baez (May 27 2020 at 19:26):

Of course they're "the same"....

Christian Williams (May 27 2020 at 19:26):

oh sorry Simon, well I'm mainly wondering "how does the adjunction really work? why does it exist?"

Christian Williams (May 27 2020 at 19:27):

I could get it by a coend calculation, but I was hoping for a bird's-eye view

Simon Willerton (May 27 2020 at 19:27):

This is the 'explanation' I gave of it in the paper.

sarahzrf (May 27 2020 at 19:27):

Simon Willerton said:

Either weak or possibly lax. You wouldn't take, say, a presheaf $f\in [C^{op},Set]$ such that $RL(f)=f$ but rather take $f\in [C^{op},Set]$ together with an isomorphism or possibly just a morphism $f\to RL(f)$

hmm, isn't the standard definition of a fixed point of an adjunction, like...

sarahzrf (May 27 2020 at 19:27):

something where the adjunction unit / counit is an iso?

Simon Willerton (May 27 2020 at 19:28):

Yes.

Simon Willerton (May 27 2020 at 19:28):

I think so.

sarahzrf (May 27 2020 at 19:29):

so that's actually incomparable with $RL(f) = f$, right?

Christian Williams (May 27 2020 at 19:29):

okay, I understand. I think when I work it out for myself then the calculation should be enough.

sarahzrf (May 27 2020 at 19:29):

like, you could have $RL(f) = f$ but the unit/counit is a non-iso endo

John Baez (May 27 2020 at 19:30):

Btw, Simon, if you're still listening: please send me your new improved talk slides!

Simon Willerton (May 27 2020 at 19:33):

I justify it as in the blog post Tai-Danae alluded to the other week. Given a matrix, thought of as a function on the cartesian product of two finite sets $M \colon A\times B\to R$ you get two adjoint functions $M^*\colon R^A \to R^B$ and $M_*\colon R^B\to R^A$ with respect to the canonical inner products on $R^A$ and $R^B$.

Simon Willerton (May 27 2020 at 19:33):

I sent you them as the talk was starting, I think.

Simon Willerton (May 27 2020 at 19:34):

Here $R=\mathbb{R}$.

John Baez (May 27 2020 at 19:34):

Okay, thanks!

Christian Williams (May 27 2020 at 19:35):

I guess the main potential for confusion is that you never think about the "opposite Yoneda embedding", the one for completion. I get that it's just the "op" of the same functor, but it's interesting.

Simon Willerton (May 27 2020 at 19:38):

For an adjunction you want $C^{op}, Set= [C,Set]^{op}(Lf, g)$ and the latter is just $C,Set$ and I think these are all just $C\times C^{op}, Set$ where P is the profunctor.

Christian Williams (May 27 2020 at 19:41):

that's nice. thanks!

John Baez (May 27 2020 at 19:42):

Of course some people think about the free completion:

sarahzrf said:

well, $[C, Set]^{\mathrm{op}}$ is the free completion

John Baez (May 27 2020 at 19:42):

We should get more used to thinking about it....

sarahzrf (May 27 2020 at 19:43):

hey, mike shulman wrote the nlab page, not me https://ncatlab.org/nlab/show/free%20completion

John Baez (May 27 2020 at 19:46):

Here are Simon's new improved slides:

http://math.ucr.edu/home/baez/mathematical/ACTUCR/Willerton_Legendre_Transform.pdf

Daniel Geisler (May 27 2020 at 22:13):

Sorry, I'd help if I could, but I don't even know who loads the different calendars.

eric brunner (May 28 2020 at 01:41):

thanks!