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Stream: theory: category theory

Topic: algebra and learning ct

John Baez (Nov 07 2020 at 16:54):

Some people, especially computer science types, might not read "do Kleisli categories always have coequalizers?" and instantly think "are quotients of free modules of rings always free?"

Dan Doel (Nov 07 2020 at 17:25):

It doesn't seem like esoteric knowledge to me, either, even though I don't think about rings all day.

John Baez (Nov 07 2020 at 17:33):

There's a difference between people who took a (typically grad-level) course on abstract algebra and those who haven't... and I have the impression that lots of people getting into category theory these days haven't.

John Baez (Nov 07 2020 at 17:34):

Some books, like Mac Lane's and Riehl's, assume you already know a fair amount of algebra before you hit category theory. I'm looking forward to an era where people learn category theory and then get lured into algebra.

John Baez (Nov 07 2020 at 17:35):

(I learned algebra for decades before messing with category theory.)

Cole Comfort (Nov 07 2020 at 17:39):

John Baez said:

There's a difference between people who took a (typically grad-level) course on abstract algebra and those who haven't... and I have the impression that lots of people getting into category theory these days haven't.

I actually met someone that learnt linear algebra from the perspective of category theory, because apparently linear algebra isn't a required course in some CS departments

John Baez (Nov 07 2020 at 17:43):

:upside_down:

John Baez (Nov 07 2020 at 17:46):

Linear algebra? I think I've heard of it: that's an example where the Kleisli category of a monad is the same as its Eilenberg-Moore algebra. Apparently that simplifies things a lot.

Dan Doel (Nov 07 2020 at 17:46):

I haven't taken a grad-level course on abstract algebra. I'm just a lowly computer science type. But I know that you can't take possibly-empty-leaf binary trees and get lists (a quotient) just by choosing the appropriate leaf type.

John Baez (Nov 07 2020 at 17:50):

Yes, there are many roads to category theory with their own examples! That's part of why it's hard to write a good introductory textbook on category theory. Do you avoid talking about all examples? Pick examples from one subject, and write Category Theory for X's? Pick examples from several subjects, and risk making everyone feel confused and ignorant?

Dan Doel (Nov 07 2020 at 17:51):

"Category Theory for Xs" doesn't go great in my experience.

Dan Doel (Nov 07 2020 at 17:52):

Only the "for mathematicians" ones are good.

Dan Doel (Nov 07 2020 at 18:03):

So maybe I would say the last approach would be good. I think the common problem with "for Xs" is that they compromise on the quality of the actual topic, not just switching examples. That might not happen if you get someone writing the usual book, but with more diverse examples.

John Baez (Nov 07 2020 at 18:06):

I'm gonna write a book on category theory, so these questions are important to me just now.

John Baez (Nov 07 2020 at 18:07):

I'm thinking of Categories for Mathematics - using categories as a way to do mathematics from the ground up, e.g. thinking of groups as special categories, etc.

John Baez (Nov 07 2020 at 18:08):

Maybe a short easy book, and then a longer deeper book.

Dan Doel (Nov 07 2020 at 18:17):

I guess I don't have a huge sample, but e.g. I think the problem with "Basic Category Theory for Computer Scientists," is that it filters the topics discussed in a very extreme way. So it's kind of like a book on some selected applications of category theory, and enough background to develop exactly those applications.

Dan Doel (Nov 07 2020 at 18:19):

It's not really a comprehensive introduction to category theory with examples from computer science.

Dan Doel (Nov 07 2020 at 18:24):

And in some ways, that's completely contrary to the point of even learning category theory. You don't want to just see topics you know presented in a different way. You want to see things people from other areas have come up with, because they might translate to something that hasn't been thought of in your discipline.

John Baez (Nov 07 2020 at 18:29):

@Bartosz Milewski, @Brendan Fong and @David Spivak are writing a book called something like Programming with Categories. I hope they overcome the problems you're talking about. Maybe you should talk to them.

Dan Doel (Nov 07 2020 at 18:33):

Yeah, I haven't looked at that yet, so I can't offer an opinion.

John Baez (Nov 07 2020 at 18:36):

I don't think it's out yet, but they have lectures on YouTube.

Dan Doel (Nov 07 2020 at 18:37):

There seems to be a draft of the book online.

John Baez (Nov 07 2020 at 18:37):

Oh, okay. Yeah, so you could offer suggestions.

Nathaniel Virgo (Nov 08 2020 at 04:24):

Personally, I think it would be pretty neat to have a book on categories that draws examples from a diverse range of subjects, but explains them all fully, without assuming much prior knowledge. That would solve a lot of the problems mentioned in this thread, and it would be a fun book to read as well, because you'd get to learn a bit about a whole load of other topics besides category theory, and if you already knew them you'd get the category-theoretic perspective. It's probably a lot more work for the author though!

সায়ন্তন রায় (Nov 08 2020 at 05:02):

John Baez said:

I'm thinking of Categories for Mathematics - using categories as a way to do mathematics from the ground up, e.g. thinking of groups as special categories, etc.

How about Categories: Chapter 0? :grinning_face_with_smiling_eyes:

John Baez (Nov 08 2020 at 21:35):

Nathaniel Virgo said:

Personally, I think it would be pretty neat to have a book on categories that draws examples from a diverse range of subjects, but explains them all fully, without assuming much prior knowledge. That would solve a lot of the problems mentioned in this thread, and it would be a fun book to read as well, because you'd get to learn a bit about a whole load of other topics besides category theory, and if you already knew them you'd get the category-theoretic perspective. It's probably a lot more work for the author though!

It would be tons of fun to write, but take a long time.

sarahzrf (Nov 09 2020 at 00:39):

John Baez said:

There's a difference between people who took a (typically grad-level) course on abstract algebra and those who haven't... and I have the impression that lots of people getting into category theory these days haven't.

John Baez said:

Some books, like Mac Lane's and Riehl's, assume you already know a fair amount of algebra before you hit category theory. I'm looking forward to an era where people learn category theory and then get lured into algebra.

hmm, this seems to be happening to me a bit

Matteo Capucci (he/him) (Nov 09 2020 at 10:51):

John Baez said:

Some books, like Mac Lane's and Riehl's, assume you already know a fair amount of algebra before you hit category theory. I'm looking forward to an era where people learn category theory and then get lured into algebra.

This was my experience when I first approached CWT. In act Algebra: Chapter 0 made me digest the algebra and the categories in a much beter way

Jules Hedges (Nov 09 2020 at 12:16):

John Baez said:

Some people, especially computer science types, might not read "do Kleisli categories always have coequalizers?" and instantly think "are quotients of free modules of rings always free?"

Indeed. I'm vaguely aware of the definition of a module, but this makes no sense to me... but I never met a kleisli category that I can't usefully think of as a category of computational processes. "Algebra for category theorists" would probably be useful to me (or at least it would be if algebra is useful, which I'm agnostic about)

sarahzrf (Nov 09 2020 at 16:45):

Jules Hedges said:

John Baez said:

Some people, especially computer science types, might not read "do Kleisli categories always have coequalizers?" and instantly think "are quotients of free modules of rings always free?"

Indeed. I'm vaguely aware of the definition of a module, but this makes no sense to me... but I never met a kleisli category that I can't usefully think of as a category of computational processes. "Algebra for category theorists" would probably be useful to me (or at least it would be if algebra is useful, which I'm agnostic about)

a left M-module for a monoid M is an algebra for the (M ⊗ -) monad! aka the writer monad

sarahzrf (Nov 09 2020 at 16:46):

this gives you exactly the ordinary notion of a left module in algebra if you note that a unital ring is a monoid in Ab (w/ tensor of abelian groups for the monoidal structure)

sarahzrf (Nov 09 2020 at 16:47):

so the kleisli category of that monad will be equivalent to the subcategory of free left M-modules

sarahzrf (Nov 09 2020 at 16:48):

well, "free" in the sense of free M-module given by an arbitrary abelian group, rather than arbitrary set

sarahzrf (Nov 09 2020 at 16:48):

however, i think you should be able to compose this with the free abelian group adjunction and recover the category of modules as monadic over Set instead of over Ab with free = ordinary notion of free

Spencer Breiner (Nov 09 2020 at 18:59):

Dan Doel said:

"Category Theory for Xs" doesn't go great in my experience.

I disagree! I think Coecke & Kissinger's quantum textbook is great for a first introduction to both CT and quantum theory. Bart Jacob's in-progress probability textbook looks good too, although I would like the string diagrams to be a bit more up front.

Spencer Breiner (Nov 09 2020 at 19:05):

John Baez said:

I'm thinking of Categories for Mathematics - using categories as a way to do mathematics from the ground up, e.g. thinking of groups as special categories, etc.

Maybe a short easy book, and then a longer deeper book.

I don't think this is a book. I think it's a whole undergraduate curriculum.

Spencer Breiner (Nov 09 2020 at 19:07):

A project like this can be seen as revisiting the founding goals of Bourbaki, to produce a modern textbook for mathematics.

Spencer Breiner (Nov 09 2020 at 19:08):

Some pain points the last time around:

• avoidance of category theory
• linear structure
• disdain for examples & pedagogy

Spencer Breiner (Nov 09 2020 at 19:16):

John Baez said:

Nathaniel Virgo said:

Personally, I think it would be pretty neat to have a book on categories that draws examples from a diverse range of subjects, but explains them all fully, without assuming much prior knowledge. That would solve a lot of the problems mentioned in this thread, and it would be a fun book to read as well, because you'd get to learn a bit about a whole load of other topics besides category theory, and if you already knew them you'd get the category-theoretic perspective. It's probably a lot more work for the author though!

It would be tons of fun to write, but take a long time.

Collaborative document seems to be the way to go.

Personally, I would like to have documents like this written at multiple levels of detail, with the same high level structure. You would have the sketch, primarily for review and reference (or experts, the "main" document at a grad-student level that adds intuition and examples, and a detailed exposition that assumes ~no background and includes basic definitions, etc.

Spencer Breiner (Nov 09 2020 at 19:20):

I think of this as a cross between a journal and classical illumination of manuscripts. The editors collectively chart the course of the text. Specific chapters/sections would be written by individuals (important stylistically, I think), with extensive criticism a la the original Bourbaki texts. Detailed expositions could be elaborated in a crowdsourced (but edited) fashion, providing teeth-cutting exercises for students and other learners.

Matteo Capucci (he/him) (Nov 09 2020 at 19:21):

This conversation is slowly converging to a description of the nlab :laughing:

Spencer Breiner (Nov 09 2020 at 19:21):

Almost, except that the nLab has no global structure

Spencer Breiner (Nov 09 2020 at 19:22):

The key is charting a good course through ~the nLab, starting from very little and building.

Matteo Capucci (he/him) (Nov 09 2020 at 19:22):

Well, if the problem is an index, we are 99% there

Spencer Breiner (Nov 09 2020 at 19:22):

nLab is almost impossible to read without ~1 year of study, minimum, I would say.

Spencer Breiner (Nov 09 2020 at 19:24):

nLab is an excellent resource, but it is an encyclopedia. It's great for reference and filling in holes, but not a great way to learn from scratch. Especially without a mathematical background.

Matteo Capucci (he/him) (Nov 09 2020 at 19:32):

There are some amazingly pedagogical articles in the nLab, with worked out examples and so on. As with every collaborative, sparse project, this style is not uniform and hardly enforced. But in the spirit of being a 'notebook', it seems skewed much more towards the pedagogical than the encyclopedical style.

Matteo Capucci (he/him) (Nov 09 2020 at 19:33):

It's not an introductory book, this is evident. You're right that you need some level of exposure to CT before engaging with it.

Matteo Capucci (he/him) (Nov 09 2020 at 19:34):

But that's the trade-off, I think. There's no encyclopedical introductory book, it would be a nonsense endeavour

sarahzrf (Nov 09 2020 at 19:38):

Spencer Breiner said:

nLab is an excellent resource, but it is an encyclopedia. It's great for reference and filling in holes, but not a great way to learn from scratch. Especially without a mathematical background.

i mostly agree that this is true, but i would like to note that there are a lot of things i have learned using the nlab as my only really comprehensive resource on the topic, so i think it has some utility as something to study from

Dan Doel (Nov 09 2020 at 19:39):

Things can be learned from a reference with sufficient background. That doesn't mean that a reference is adequate for acquiring that background.

sarahzrf (Nov 09 2020 at 19:41):

yeah, i should be careful not to leave out the caveat that i would probably not have been able to use the nlab the way i have if i hadnt had plenty of people with relevant knowledge to help me put pieces together and supply background & stuff

sarahzrf (Nov 09 2020 at 19:41):

& additional context added by other sources that ive looked at, etc

Dan Doel (Nov 09 2020 at 19:41):

I think one of the primary problems with nlab, for instance, is that it doesn't pick any of the fundamental ways of characterizing things categorically. So you just go in loops if you don't know one of those already. "A limit is an end is a right adjoint is a right kan extension is a limit ...".

Dan Doel (Nov 09 2020 at 19:42):

Books pick one of those to focus on and then build the others.

Shea Levy (Nov 09 2020 at 19:42):

This has been a positive about nlab at times for me. Instead of a cycle, it turns into a spiral over time. But agreed it's not a great fit for a single-pass book

Dan Doel (Nov 09 2020 at 19:44):

It's good if you do know one of those already, too, because then you can bottom out on the one you already know, instead of the nlab picking the one you don't know.

Dan Doel (Nov 09 2020 at 19:47):

It does have some obscure loops even for people who know a lot, though, I think. Like this. That's like a web of a dozen articles that are just like, "a supermanifold is like a manifold in geometry except in supergeometry".

Dan Doel (Nov 09 2020 at 19:49):

So, you can use that to experience what the nlab is like if you don't know anything yet. :)

John Baez (Nov 09 2020 at 22:39):

Spencer Breiner said:

John Baez said:

Nathaniel Virgo said:

Personally, I think it would be pretty neat to have a book on categories that draws examples from a diverse range of subjects, but explains them all fully, without assuming much prior knowledge.

It would be tons of fun to write, but take a long time.

Collaborative document seems to be the way to go.

John Baez (Nov 09 2020 at 22:46):

A neo-Bourbaki-esque collaborative project developing all of math with the help of categories would be great to see... but I don't see anyone out there now with that level of energy for writing except the folks working on the nLab, the HoTT crowd, and the Stacks project. (I guess Jacob Lurie counts as a one-man army.)

John Baez (Nov 09 2020 at 22:49):

Even apart from the energy required, it would be hard to tackle a serious neo-Bourbaki-esque project right now because the best approach to mathematics is in such flux now. For example, a bunch of smart people think everything that used to be done with categories should now be done with homotopy type theory or $(\infty,1)$-category theory - or at least, as much as possible.

John Baez (Nov 09 2020 at 22:50):

So, for example, a massive introduction to topos theory starts seeming ever so slightly obsolete, now that so many people are working on $(\infty,1)$-topoi.

John Baez (Nov 09 2020 at 22:51):

And then of course there are huge chunks of geometry and analysis that currently use just a small amount of category theory.

John Baez (Nov 10 2020 at 02:13):

So one might want to carefully think about a "multi-layered" approach. Sounds like a lot of work!

Jens Hemelaer (Nov 10 2020 at 10:17):

One of the things that I miss is a "recipe" to translate results from the $(\infty,1)$-world to the world of 1-categories.

For example, Lurie in "Higher Topos Theory" has results about $(\infty,1)$-toposes, whose statement can be easily translated to a statement for 1-toposes, but, as far as I can see, there is no way to know that it holds for 1-toposes as well (other than translating the proof).

The other direction, a recipe of translating 1-categorical results to $(\infty,1)$-categorical results, seems equally difficult.

Reid Barton (Nov 10 2020 at 12:16):

For topos theory in particular there is a difficulty because a 1-topos (e.g. Set) is not an $(\infty,1)$-topos and from an $(\infty, 1)$-point of view it's not really clear what a 1-topos is. In particular an $(\infty,1)$-topos has a descent property which holds for all colimits, and also makes sense in a 1-topos but is false in general there (https://ncatlab.org/nlab/show/van%20Kampen%20colimit#examples).

In most other cases 1-categorical notions are instances of the corresponding $(\infty,1)$-categorical ones, and the 1-categories can be thought of as the Set-enriched $(\infty,1)$-categories. (Though perhaps here we could call it "Set-depleted"; note that a 1-category is an $(\infty,1)$-category enriched in 0-truncated spaces (sets) and a poset is an $(\infty,1)$-category enriched in (-1)-truncated spaces (truth values).) Set is characterized as a presentable $(\infty,1)$-category by the single relation which is the non-van Kampen pushout square in the link above. (Compare: A category is a poset if and only if the fold map $X \amalg X \to X$ is an isomorphism for every $X$.)

Going the other direction, from theorems about 1-categories to theorems about $(\infty,1)$-categories, is the hard part, since truncation loses information. For example, homotopy limits cannot be defined as limits in the homotopy category.

Jens Hemelaer (Nov 10 2020 at 13:42):

Maybe the translation from $(\infty,1)$-categorical results to 1-categorical ones is easier than I think then, and the issue is mostly with toposes or going in the other direction. I'll start a topic about it, because I don't want to derail too much the conversation here about the pedagogical aspects.

Joe Moeller (Nov 10 2020 at 22:15):

It's been an ambition of mine since I was an undergrad to write a textbook which covers all pure math undergrad material in a systematic way. This is clearly way too ambitious and definitely wouldn't come out very good. I then considered writing an intro to category theory which assumes very little. I'm now of the opinion that a much better approach is to write a bunch of smaller books which are "Intro to X", but which simply introduces categorical language as it goes. It has the sneaky side-effect of side-stepping the ct haters. I also don't like the idea of giant tomes. They could only possibly be off-putting to beginners.

John Baez (Nov 10 2020 at 22:20):

@Bob Coecke and Joshua Tan are editing a book series, "Reasoning with Categories", to be published with Cambridge University Press. It's meant to be a series of 80-120 page monographs introducing specific applications of CT in different subjects (or introducing different subjects through the lens of CT.

John Baez (Nov 10 2020 at 22:22):

So far the only book I know in this series is Noson Yanofsky's Theoretical Computer Science for the Working Category Theorist.

John Baez (Nov 10 2020 at 22:23):

If anyone here wants to write such a book, contact Bob!

Jules Hedges (Nov 10 2020 at 22:38):

A couple of years ago I did a bunch of work for a book on open games for that series, before deciding that writing a book is too hard

Chetan Vuppulury (Nov 11 2020 at 14:40):

I have long been looking for a book on 'Analysis for the Working Category Theorist' that maybe consists of introductions to various branches of analysis. At least some of analysis has categorical flavour (Abstract harmonic Analysis, C* algebras). Maybe @Tobias Fritz 's work on Markov categories could also be introduced. Does anyone know of such a work?

Fawzi Hreiki (Nov 11 2020 at 14:43):

Helemskii's 'Lectures and Exercises on Functional Analysis' does a little in that direction

Simon Burton (Nov 12 2020 at 12:18):

Spencer Breiner said:

Dan Doel said:

"Category Theory for Xs" doesn't go great in my experience.

I disagree! I think Coecke & Kissinger's quantum textbook is great for a first introduction to both CT and quantum theory. Bart Jacob's in-progress probability textbook looks good too, although I would like the string diagrams to be a bit more up front.

Does anyone know where to find this probability textbook ?

Nathaniel Virgo (Nov 12 2020 at 13:05):

I think it's this one: https://www.cs.ru.nl/B.Jacobs/PAPERS/ProbabilisticReasoning.pdf