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Eric M Downes (May 16 2024 at 16:55):One Tuesday May 29 JS Lemay will be giving a talk "tour around the world of differential categories"
Here is an abstract:
"""
The theory of differential categories uses category theory to provide the abstract foundations of differential calculus in both mathematics and computer science. In this talk, I will provide a tour on the world of the differential categories. We will visit the four chapters of differential categories:
Differential categories, which axiomatize the algebraic foundations of differentiation.
Cartesian differential categories, which axiomatize differential calculus over Euclidean spaces.
Differential restriction categories, which axiomatize differential calculus over open spaces.
Tangent categories, which axiomatize differential calculus over smooth manifolds.
We will talk about the famous "map of differential categories", and also the history and applications of differential categories.
"""
Jean-Baptiste Vienney (May 28 2024 at 18:16)::ram:: This is tomorrow!
Morgan Rogers (he/him) (May 29 2024 at 09:04):Is that... a raminder?
Ralph Sarkis (May 29 2024 at 09:39):It is at (right?) to make the conversion for people.
Eric M Downes (May 29 2024 at 09:40):I don't know what timezone you're in Ralph. It's 9 AM CDT, 1400 GMT.
Eric M Downes (May 29 2024 at 09:41):Gawd I hope the schedule from researchseminars.org don't all say 9 am irrespective of someone's timezone O_0
Bryce Clarke (May 29 2024 at 09:45):Eric M Downes said:
Gawd I hope the schedule from researchseminars.org don't all say 9 am irrespective of someone's timezone O_0
Researchseminars automatically states the time in the reader's own timezone.
Eric M Downes (May 29 2024 at 09:47):Thanks! Phew, that's what I thought, but I realized I didn't actually check.
Jean-Baptiste Vienney (May 29 2024 at 09:47):Ralph Sarkis said:
It is at (right?) to make the conversion for people.
The time in Ralph’s message should appear in your time zone too.
Eric M Downes (May 29 2024 at 09:49):what whitchcraft is this!! (Zulipmancy, I suppose.) Ok see everybody in ~ 4 hrs. Sorry I didn't realize you could use a cool macro when creating the topic. Will do that in the future.
Eric M Downes (May 29 2024 at 15:48):Ok! Talk recording uploaded in the usual place @JS PL (he/him) if would like to make your slides available, I will add those as well! DM me.
Eric M Downes (May 29 2024 at 16:30):Ok! Slides now in JSPL's folder.
JS PL (he/him) (May 29 2024 at 17:56):Thank you for the invitation to speak and thank you to all who attended and listened. I hope enjoyed and found it interesting and learned something!
Todd Trimble (May 29 2024 at 20:57):Thanks; it was a nice and helpful overview!
I had a boring technical question. There's a paper by Carboni on bicategories of partial maps, similar in spirit to bicategories of relations in the sense of Carboni-Walters in their Cartesian Bicategories paper. Can you comment on what relation there is between that and restriction categories?
JS PL (he/him) (May 30 2024 at 07:28):Todd Trimble said:
Can you comment on what relation there is between that and restriction categories?
Yes there's definitely a connection! I am less familiar with Carboni's notion of bicategory of partials maps, but my understanding and recollection is that restriction categories are more general and I believe give rise to a Carboni's notion of a bicategory of partial maps. However I'm not 100% familiar with the exact connection. Restriction categories I, II, and III do reference Carboni and Carboni-Walters papers, so a deeper explanation might be found there. Or others like @Chad Nester or @Cole Comfort might have a better understanding of the connection between the two concepts.
Chad Nester (May 30 2024 at 07:48):If I recall correctly, then Carboni's bicategories of partial maps are very close to what we have been calling "discrete cartesian restriction categories".
Chad Nester (May 30 2024 at 08:11):... Checking the paper, I'm pretty sure that they're the same thing. We need the "bicategories of partial maps" from section 2 though, not the "bicategories equipped with a structure of a bicategory of partial maps" of section 1, which are missing some axioms.
Chad Nester (May 30 2024 at 08:15):These categories play a pretty big role in my doctoral thesis, so if you'd like more details there's a good chance they're in it. Also happy to talk more about this here :)
Chad Nester (May 30 2024 at 08:17):One day "someone" should really write a (series of) survey articles covering the many different notions of category of partial maps and how they're related.
Eric M Downes (May 30 2024 at 08:18):Chad Nester said:
These categories play a pretty big role in my doctoral thesis, so if you'd like more details there's a good chance they're in it. Also happy to talk more about this here :)
Perhaps you'd also be happy to talk more about this... over zoom at our seminar this fall??? :)
Chad Nester (May 30 2024 at 08:19):Sure!
Damiano Mazza (May 30 2024 at 08:24):This paper by @Tobias Fritz, Fabio Gadducci, @Davide Trotta, and @Andrea Corradini seems to say something relevant.
The paper introduces oplax cartesian cartegories as certain pre-order-enriched [[gs-monoidal categories]]. These are a relaxed version of Carboni and Walter's bicategories of relations and, if I am not mistaken, Carboni's bicategories of partial maps are exactly oplax cartesian categories in which the duplicator is natural (rather than oplax natural).
In Remark 2.15 (p.9), the authors say that "restriction categories with restriction products correspond exactly to gs-monoidal categories whose duplicator is natural". So the relationship between restriction categories with restriction products and bicategories of partial maps seems to be that the latter are a pre-order-enriched version of the former.
Damiano Mazza (May 30 2024 at 08:31):More concisely: restriction categories with restriction products are exactly the same thing as gs-monoidal categories with natural duplicator. If you take such categories, add pre-order-enrichment and ask that the eraser is oplax natural, you obtain Carboni's bicategories of partial maps.
Chad Nester (May 30 2024 at 08:32):I'm not sure that this is true. Where does the right adjoint to the duplicator come from?
Damiano Mazza (May 30 2024 at 08:32):There is no right adjoint in oplax cartesian categories.
Chad Nester (May 30 2024 at 08:33):But I'm pretty sure there is one in Carboni's bicategories of partial maps, no?
Damiano Mazza (May 30 2024 at 08:34):In bicategories of relations, yes, in bicategories of partial maps, I don't think so.
Chad Nester (May 30 2024 at 08:34):Okay, well the definition says there is, so I'm not sure what we're talking about. (this paper)
Damiano Mazza (May 30 2024 at 08:35):Ok then I'm misremembering the definition :-D
Chad Nester (May 30 2024 at 08:37):Also any restriction category is poset-enriched, and if it has finite restriction products then the eraser is oplax natural with respect to this ordering.
Damiano Mazza (May 30 2024 at 08:39):Ok then I was also misremembering the definition of restriction category. I didn't know that it implied poset enrichment!
Chad Nester (May 30 2024 at 08:39):We really need that survey paper :)
Damiano Mazza (May 30 2024 at 08:41):The authors of the paper I mentioned maybe also missed this. They never seem to say that gs-monoidal categories in which the duplicator is natural are automatically oplax cartesian (that is, poset-enriched and with oplax eraser).
Damiano Mazza (May 30 2024 at 08:47):Chad Nester said:
... Checking the paper, I'm pretty sure that they're the same thing. We need the "bicategories of partial maps" from section 2 though, not the "bicategories equipped with a structure of a bicategory of partial maps" of section 1, which are missing some axioms.
Ok, so now I see what's going on. We weren't talking about the same thing. I was referring to Carboni's Definition 1.1. There is no right adjoint to the duplicator. This is exactly what the authors of the other paper call an oplax cartesian category with natural duplicator.
Damiano Mazza (May 30 2024 at 08:48):So what I said above is true, you just need to read "bicategory equipped with a structure of a bicategory of partial maps" everywhere I wrote "bicategory of partial maps". Right?
Damiano Mazza (May 30 2024 at 08:53):But wait, you said that every gs-monoidal category with natural duplicator is automatically oplax cartesian. So:
are all the same thing. Is that right?
Cole Comfort (May 30 2024 at 12:13):To give an overview of what Chad was talking about... There are various notions of categories with some kind of partiality, and there is a nice story to tell connecting some of them. I don't know the full picture.
The notion of products present in Cartesian categories can be generalised in several ways such that different universal properties hold. A Cartesian bicategory is the Span-like generalisation (which I won't discuss, because it isn't poset enriched, unlike the others); a Cartesian bicategory of relations is the "Rel-like" nondeterministic version; a discrete Cartesian restriction category is the partial version; and a discrete inverse category is the partially invertible version.
Take a Cartesian bicategory of relations, which is equipped with a distinguished extraspecial Frobenius algebra on each object, compatible with the tensor product. Then as a matter of notation, to avoid confusion, by comonoid, monoid, Frobenius algebra, multiplication, comultiplication etc., I mean those coming from the Cartesian bicategory of relations structure. By asking that maps be strictly/laxly/oplaxly natural with respect to the different components of the Frobenius algebras, you get these different weakenings of a Cartesian bicategory of relations.
The subcategory of comonoid homorphisms is Cartesian, the subcategory of comultiplication homorphisms is a discrete Cartesian restriction category, and the category of semi-Frobenius (non unital, non counital Frobenius algebra) homomorphisms is a discrete inverse category.
One part of this correspondence extends to an equivalence of categories between the category of discrete inverse categories and the category of discrete Cartesian restriction categories. This is witnessed on objects, on the one hand, by sending a discrete Cartesian restriction category to its subcategory of semi-Frobenius homorphisms; and on the other hand, by freely adding counits to the comultiplication (non counital comonoid) structure of a discrete inverse category.
I would have liked to have shown that there is also an equivalence between the category of discrete Cartesian restriction categories and the category of Cartesian bicategories of relations: on the one hand, by looking at comultiplication homomorphisms of Cartesian bicategory of relations; and on the other by taking a discrete Cartesian restriction category and freely adding units to the multiplications (non unital monoid) such that they are also made to be sections of the counits. However, I have a feeling that this is false in general, and it seems that ones into trouble with certain non-split idempotents. If my suspicion is correct, then Cartesian bicategories of relations would be strictly less expressive than discrete Cartesian restriction categories.
Cole Comfort (May 30 2024 at 12:18):To orient oneself:
Selection_149.png
Chad Nester (May 30 2024 at 13:55):Damiano Mazza said:
But wait, you said that every gs-monoidal category with natural duplicator is automatically oplax cartesian. So:
- Carboni's "bicategories equipped with a structure of a bicategory of partial maps";
- gs-monoidal categories with natural duplicator;
- restriction categories with restriction products
are all the same thing. Is that right?
I think so!
Chad Nester (May 30 2024 at 13:58):@Cole Comfort I think you need to work with the "regular restriction categories" of Cockett, Hofstra and Guo to get an equivalence with cartesian bicategories of relations. If discrete cartesian restriction categories are equivalent to cartesian bicategories of relations in this sense, then we also have that categories with finite limits and regular categories are equivalent (which they should not be).
Chad Nester (May 30 2024 at 14:01):... In particular it's straightforward to show that the deterministic arrows in a cartesian bicategory of relations form a regular restriction category.
Chad Nester (May 30 2024 at 14:02):I think that actually your suspicion is the wrong way around!
Cole Comfort (May 30 2024 at 14:13):@Chad Nester are regular restriction categories specific kinds of discrete Cartesian restriction categories?
What I meant to say, is that I believe that there is an equivalence between the category of Cartesian bicategories of relations and some subcategory of the category of discrete Cartesian restriction categories.
If what you say is true, it would also be interesting to know what a "regular discrete inverse category" is, in order to come closer to a more complete picture of what is going on.
Chad Nester (May 30 2024 at 14:18):A regular restriction category is a "discrete cartesian range category in which every partial monic has a partial inverse". This takes some unpacking, but yes, they are discrete cartesian restriction categories with more stuff. (from this)
Chad Nester (May 30 2024 at 14:19):I think these are probably the ones you want.
Chad Nester (May 30 2024 at 14:20):I wonder how ranges work in discrete inverse categories. I guess it's probably similar. Your matrix of structured categories would gain another dimension :thinking:
Cole Comfort (May 30 2024 at 15:51):Chad Nester
There are also Markov categories and partial Markov categories and copy-delete categories which could also be added (perhaps as another dimension because I don't think they are a priori enriched in anything other than set)
Morgan Rogers (he/him) (May 30 2024 at 16:27):I hope that some of these classes will acquire more inspiring names once we have more of them around :sweat_smile:
Robin Piedeleu (May 31 2024 at 06:59):Cole Comfort said:
To give an overview of what Chad was talking about... There are various notions of categories with some kind of partiality, and there is a nice story to tell connecting some of them. I don't know the full picture.
The notion of products present in Cartesian categories can be generalised in several ways such that different universal properties hold. A Cartesian bicategory is the Span-like generalisation (which I won't discuss, because it isn't poset enriched, unlike the others); a Cartesian bicategory of relations is the "Rel-like" nondeterministic version; a discrete Cartesian restriction category is the partial version; and a discrete inverse category is the partially invertible version.
Thank you for the fantastic overview of these fantastic beasts! I just wanted to comment on your side-point here and add to the overall picture: a Cartesian bicategory - not necessarily of relations - in the sense of Carboni & Walters' Cartesian Bicategories I is still poset-enriched if I'm not mistaken. Without going to the non-locally posetal bicategory of Spans (that's in Cartesian Bicategories II iirc), there are interesting examples of such categories, where the diagonal and projections have right adjoints, but together these do not form a Frobenius algebra. One paradigmatic examples is the category of Boolean-enriched profunctors aka monotone relations (aka weakening relations, aka...? It has many names in the literature since various people have rediscovered its use independently): its objects are (pre-)ordered sets and morphisms are relations that preserve the order in a certain sense. In this category, the objects for which the structure maps do form a Frobenius algebra are precisely those whose order relation is equality, i.e., sets! So Rel (the category of ordinary relations) lives inside of it, but arbitrary objects need not be discrete in the sense of Carboni & Walters. It is still compact closed however, and the dual of an object is not itself, but , the same set equipped with the opposite order relation.
Cole Comfort (May 31 2024 at 11:18):@Robin Piedeleu Like @Morgan Rogers (he/him) said, these names are so confusing... there needs to be a dictionary
Robin Piedeleu (May 31 2024 at 11:24):Agreed! I just wanted to point out the existence of another level in this hierarchy: the poset-enriched case where the copy and delete operations do have right adjoints, but these four do not form a Frobenius algebra (and not just because I love monotone relations and felt that they deserved a mention)