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sarahzrf (Mar 23 2020 at 22:12):@Ben Steffan posted on twitter asking about an "algebraic" study of categories & i got into a whole convo w/ him about it
Screen-Shot-2020-03-23-at-18.09.47.png
see here https://twitter.com/BiCapitalize/status/1242196352362307584
& now i wanna know if anyone else has a take on this
@sarah_zrf Now, I'm definitely not an expert on homological algebra either but the little I have seen was... not that
- Ben Steffan (@BiCapitalize) Nathanael Arkor (Mar 23 2020 at 22:15):I think an algebraic study of categories would really be in the style of the theory of categories as a generalised algebraic theory or essentially algebraic theory
Evan Patterson (Mar 23 2020 at 22:15):IMO, the whole study of category theory is "algebraic", but the term "categorical algebra" is sometimes used for, to quote the nLab, "those aspects of categorical and category-like constructions which are in the spirit of pure algebra."
Nathanael Arkor (Mar 23 2020 at 22:15):for example: https://www.mta.ca/uploadedFiles/Community/Bios/Geoff_Cruttwell/ams2014CruttwellCountingFiniteCats.pdf
Nathanael Arkor (Mar 23 2020 at 22:16):@Evan Patterson: that's true, but I don't think that's what Ben Steffan is referring to in that conversation
Nathanael Arkor (Mar 23 2020 at 22:16):it's more in the spirit of "a category is an algebraic structure โย what general theorems can we develop about, say, its combinatorics"
John Baez (Mar 23 2020 at 22:18):In my very slowly emerging textbook on category theory I distinguish between "categories of algebraic gadgets" and "categories as algebraic gadgets". To really study the latter, I think we need 2-category theory... since there's not just a category of categories: there's a 2-category of them.
sarahzrf (Mar 23 2020 at 22:19):other results besides giraud that feel to me like they're "algebraic" are stuff like the monadicity theorem, i think
sarahzrf (Mar 23 2020 at 22:19):or the adjoint functor theorem
Evan Patterson (Mar 23 2020 at 22:19):@Nathanael Arkor, what do you think of the study of finitely presented categories, like the theory of graphs, as in texts about presheaf toposes like the one by Reyes, Reyes and Zolfaghari? To me, this is the analogue of studying finitely presented groups as is typical in algebra.
sarahzrf (Mar 23 2020 at 22:20):in another place the other day i think i referred to the way algebra treats its objects of study as "anatomical" and i feel like that's sort of characterizing to me
sarahzrf (Mar 23 2020 at 22:21):like you're doing a dissection or autopsy on a ring
sarahzrf (Mar 23 2020 at 22:21):really pulling it apart into pieces and labelling them
Nathanael Arkor (Mar 23 2020 at 22:24):@Evan Patterson: ah, you're right, that does have a similar flavour
Nathanael Arkor (Mar 23 2020 at 22:25):maybe the fact that categories are a "dependent theory" means their study looks inherently different from algebraic structures
Nathanael Arkor (Mar 23 2020 at 22:26):and so most of pure category theory is really the same sort of study
Ben Steffan (Mar 23 2020 at 22:39):@sarahzrf I feel like that is moving in the direction I was envisioning
sarahzrf (Mar 23 2020 at 22:40):which one? my 'anatomical' description? monadicity?
Ben Steffan (Mar 23 2020 at 22:40):Anatomicality
Matteo Capucci (he/him) (Mar 24 2020 at 13:18):John Baez said:
In my very slowly emerging textbook on category theory I distinguish between "categories of algebraic gadgets" and "categories as algebraic gadgets". To really study the latter, I think we need 2-category theory... since there's not just a category of categories: there's a 2-category of them.
Exactly, this is called formal category theory sometimes. See https://twitter.com/mattecapu/status/1242439983241277441?s=20
@BiCapitalize How come nobody cited the Australian school yet? Afaik it's concerned with "formal CT", i.e. what you might call the algebraic study of categories. Mostly this means trying to come up with the things you use in CAT (Yoneda, monads, adjunctions, ...) in any 2-category
- ๐ฎ๐น๐ช๐บ ๐๐๐ฅ๐ฅ๐๐ โ๐๐ก๐ฆ๐๐๐ (@mattecapu) nadia esquivel mรกrquez (Mar 26 2020 at 15:47):Besides classification, characterizations of (nice) objects satisfying some (nice) properties are the central goals "algebraic" study. Besides Giraud's theorem which @sarahzrf already mentioned i can't think of any particular results in ct.
sarahzrf (Mar 26 2020 at 15:48):yeah thats the kind of thing i had in mind when i brought it up :)
sarahzrf (Mar 26 2020 at 15:48):it's also what i was thinking of when i mentioned the monadicity theorem
sarahzrf (Mar 26 2020 at 15:48):& adjoint functor theorem
Matteo Capucci (he/him) (Mar 26 2020 at 18:29):Oh, I see. So you mean like classification results?
Pierre Cagne (Mar 26 2020 at 21:25):Not sure if that is what you're after, but the category Cat of (small) categories is the category of algebras of a monad on the category of graphs. This monad is algebraic in a very precise way: it is a monad generated by a "Lawvere theory with arities", a fairly natural generalization of Lawvere theories (aka universal algebra) where operations are not restricted to have natural numbers as arities (boring!) but they are allowed to have "shape" arities. The reference paper is https://arxiv.org/pdf/1101.3064.pdf (it is a little stiff though).
Nathanael Arkor (Mar 26 2020 at 21:27):Monads and theories (https://arxiv.org/abs/1805.04346) is a nicely-written generalisation
Nathanael Arkor (Mar 26 2020 at 21:28):though another way to describe the theory is as a 2-Lawvere theory