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Stream: learning: questions

Topic: motivation for accessible categories

Leopold Schlicht (Sep 09 2021 at 11:13):

What was/is the motivation for accessible categories? What are applications of them? For instance, why did Grothendieck and Verdier talk about accessible functors in SGA4?

Zhen Lin Low (Sep 09 2021 at 11:24):

Accessible categories are not-necessarily small categories that are generated by a small category in a particularly nice, controlled way. It's a way of addressing some size problems.

Leopold Schlicht (Sep 09 2021 at 11:29):

Nice! What would be a concrete example of a classical such size problem addressed by accessible categories? (That one doesn't have to check the technical conditions in the adjoint functor theorems?)

Zhen Lin Low (Sep 09 2021 at 11:31):

Since you mentioned the adjoint functor theorem: any accessible functor between locally presentable categories is a left adjoint iff it preserves colimits, and any accessible functor between locally presentable categories is a right adjoint iff it preserves limits.

Leopold Schlicht (Sep 09 2021 at 11:45):

Cool! So that's all, accessible categories are just about allowing annoying conditions to be omitted?

Zhen Lin Low (Sep 09 2021 at 11:46):

No... I would say that's not the point at all...

Zhen Lin Low (Sep 09 2021 at 11:49):

There's no free lunch – call it conservation of annoyingness if you like – but packaging technical conditions into a definition doesn't make it go away!

Leopold Schlicht (Sep 09 2021 at 11:52):

Zhen Lin Low said:

No... I would say that's not the point at all...

Then, what's the point of accessible categories? :grinning_face_with_smiling_eyes:

Zhen Lin Low (Sep 09 2021 at 11:55):

Why don't you go read a textbook and find out for yourself?

Jules Hedges (Sep 09 2021 at 11:58):

Zhen Lin Low said:

Why don't you go read a textbook and find out for yourself?

Allow me to politely remind you the tagline of this channel. I'm considering it against the Rules to say this to somebody here, any question you don't want to answer just leave for somebody else

Zhen Lin Low (Sep 09 2021 at 12:05):

Fine. I'll ignore questions I don't want to answer even if they are addressed to me directly.

Jules Hedges (Sep 09 2021 at 12:39):

Zhen Lin Low said:

Fine. I'll ignore questions I don't want to answer even if they are addressed to me directly.

I won't say much more as a moderator, we try to keep a light touch, the goal in this case is to keep things friendly and to prevent this from becoming a less good version of MathOverflow (we try to occupy a different niche, since MO has its own niche very well covered). Speaking as a human rather than as a moderator, the best thing is to politely disengage yourself from the thread, but just ignoring it is pretty much fine too, somebody else will probably pick it up

Morgan Rogers (he/him) (Sep 09 2021 at 12:39):

Zhen Lin Low said:

Fine. I'll ignore questions I don't want to answer even if they are addressed to me directly.

You may have been speaking facetiously, but doing that does actually work (without seeming rude!) in this format.

Morgan Rogers (he/him) (Sep 09 2021 at 12:41):

Re accessible categories: most of category theory is about studying categories which are constructed in a particular way, or which are "just nice enough" for some phenomenon or other to work in/on them. One motivation for accessible categories comes from studying models of sketches, another is understanding categories with certain classes of colimits. But these are things you could have learned from just reading the nLab page @Leopold Schlicht; what deeper insight are you looking for here?

Mateo Carmona (Sep 09 2021 at 13:45):

In case you haven't seen it yet, take a look at chapter XVIII of Dérivateurs, where Grothendieck revisits the question in terms of "pseudo-topos". You can ask Maltsiniotis for the other sections if you get caught. https://webusers.imj-prg.fr/~georges.maltsiniotis/groth/Derivateurs.html G says some words about it at the beginning of this letter https://webusers.imj-prg.fr/~georges.maltsiniotis/groth/ps/lettreder.pdf

John Baez (Sep 09 2021 at 14:32):

Leopold Schlicht said:

Cool! So that's all, accessible categories are just about allowing annoying conditions to be omitted?

If you left out the word "just" I would say yes, this is one of the things they're good for.

John Baez (Sep 09 2021 at 14:32):

But these are things you could have learned from just reading the nLab page @Leopold Schlicht; what deeper insight are you looking for here?

I again find a bit of "question-shaming" is at work here. I don't want people to feel they need to study the nLab before they ask a question here.

John Baez (Sep 09 2021 at 14:33):

I think Leopold's original question was great: what are accessible good for? One can ask this without knowing what particular "deeper insight" one is seeking.

John Baez (Sep 09 2021 at 14:36):

Come to think of it, when I'm looking for deeper insight into something, I never know what deeper insight I'm looking for until I've got it!

John Baez (Sep 09 2021 at 14:41):

So I think you should go on asking such questions, Leopold - and anyone who doesn't like them should remember that silence is often more useful than a grumpy reply.

John Baez (Sep 09 2021 at 14:44):

Personally I've found locally presentable categories to be easier to like than accessible categories... but locally presentable categories are the same as cocomplete accessible categories, so when you're trying to use locally presentable categories you need to read results about accessible categories.

Morgan Rogers (he/him) (Sep 09 2021 at 14:49):

John Baez said:

But these are things you could have learned from just reading the nLab page @Leopold Schlicht; what deeper insight are you looking for here?

I again find a bit of "question-shaming" is at work here. I don't want people to feel they need to study the nLab before they ask a question here.

Quite the opposite: I was admitting to the fact that, if Leopold was asking here after going to the nLab, then I wasn't really providing them with anything new! I also am encouraging more specific questions; is @Leopold Schlicht asking us to "sell them" accessible categories, or asking for more specific applications of them, or something else?

John Baez (Sep 09 2021 at 14:51):

Here's what I tend to remember about locally presentable categories:

1) They have a somewhat technical definition, but they're the same as categories of models of limits theories - which is nice, because this includes tons of categories I care about.

2) The adjoint functor theorem gets a lot simpler for locally presentable categories.

John Baez (Sep 09 2021 at 14:54):

Morgan wrote:

I was admitting to the fact that, if Leopold was asking here after going to the nLab, then I wasn't really providing them with anything new!

Interesting. I read it the opposite way, since often people ask for help after looking at the nLab and not getting the point of what's there. (Often the information is there but quite hard to digest if you don't know the point of it all.)

Fawzi Hreiki (Sep 09 2021 at 15:07):

A good motivation is just that so many of the large categories of structures which naturally occur are accessible

Fawzi Hreiki (Sep 09 2021 at 15:08):

So it just makes sense to see what the axioms alone imply

Fawzi Hreiki (Sep 09 2021 at 15:09):

More theoretically, there’s also the fact that a category is accessible if and only if it’s sketchable. So accessibility is a nice categorical axiomatisation of ‘presentable by axioms’ (roughly)

Mike Shulman (Sep 09 2021 at 16:01):

Here's some relevant discussion at mathoverflow (starting in the $\infty$-case, but the points are mostly the same).

John Baez (Sep 09 2021 at 16:34):

Fawzi Hreiki said:

More theoretically, there’s also the fact that a category is accessible if and only if it’s sketchable. So accessibility is a nice categorical axiomatisation of ‘presentable by axioms’ (roughly)

I think a category is locally presentable iff it's equivalent to the category of models of a limits sketch. Does "sketchable" mean something else?

John Baez (Sep 09 2021 at 16:35):

Oh, it means sketchable using limits and colimits.

John Baez (Sep 09 2021 at 16:36):

Now I'll finally start to love accessible categories!

Peter Arndt (Sep 09 2021 at 17:16):

Here is a nice statement that is true in every locally presentable category C: If X and Y are types of structures that can be specified using only limits (i.e. by limit sketches, e.g. groups, rings, lie algebras, internal categories) then an X in the categories of Ys in C is the same as a Y in the category of Xs in C. E.g. a category object in group objects in Lie algebras is the same as a group object in category objects in Lie algebras. This follows from the commutation of limits and the fact that C itself is a category of models (in Set) of a limit sketch.
From the discussion here you can also extract the corresponding statements in general accessible categories and with more general sketches.

Peter Arndt (Sep 09 2021 at 17:17):

Another one: An accessible category is complete if and only if it is cocomplete. This one, in its $\infty$-categorical version, really comes to the rescue when you treat Lie algebras...

Todd Trimble (Dec 28 2021 at 00:00):

Returning to this thread: this was touched on a bit already, but I think it's useful to think of the theory of locally presentable categories as "categorical model theory", as the famous book by Makkai and Parë puts in in the title. Actually, what it is is a beautiful and satisfying interplay and duality between syntax (generalizing something akin to Lawvere theories) and semantics (generalizing models of such theories).

Todd Trimble (Dec 28 2021 at 00:01):

One good way to get started is with Gabriel-Ulmer duality, which is very striking. In the first place, this concerns "finite-limits logic" and "finite limits theories". So you have to imagine the sorts of internal structures in categories that can be expressed using the language of finite limits. Group objects in a category would be an example -- to define those those you just need products in the category. Category objects would be another example -- there you need pullbacks. Poset objects would be another example.

Todd Trimble (Dec 28 2021 at 00:01):

The classical case would be where the background category is the category of sets. Internal posets in $\mathsf{Set}$ are just... posets. Category objects in $\mathsf{Set}$ are just small categories. And so on. The category of groups, the category of small categories, the category of posets -- these are all examples of what are called locally finitely presentable categories.

Todd Trimble (Dec 28 2021 at 00:02):

Slightly more precisely, one way of defining a locally finitely presentable category is that it is a category $C$ equivalent to a category of finite-limit preserving functors

$M: T \to \mathsf{Set}$

from a small category $T$ that has finite limits (is finitely complete). Such a $T$ can be thought of as a "finite limits theory", and the finitely continuous functors $M: T \to \mathsf{Set}$, also called "left exact" or "lex" functors, can be thought of as "models" of $T$. Very much in the spirit of Lawvere theories.

Todd Trimble (Dec 28 2021 at 00:02):

Now, this is not really an intrinsic definition of an LFP category $C$ -- we have to be told there is some small finite-limits theory $T$ out there that $C$ can be presented in terms of. What is quite amazing though is that the $T$ in question is essentially unique and can be extracted from $C$. Namely, if we are told $C$ is LFP, then define an object $c$ of $C$ to be "finitely presentable" if the representable functor

$C(c, -): C \to \mathsf{Set}$

preserves filtered colimits. Bring in the nice fact that in $\mathsf{Set}$, filtered colimits commute with finite limits. It follows that such representable functors are closed under finite limits; consequently, the representing objects themselves, aka finitely presentable objects, are closed under finite colimits. The theory $T$ we are after is the category opposite to the full subcategory of finitely presentable objects. (Somewhat akin to the way in which a Lawvere theory for a category of algebras is opposite to the category of finitely generated free algebras.)

Todd Trimble (Dec 28 2021 at 00:03):

For example, the canonical finite-limits theory for the theory of groups is the category opposite to the category of finitely presented groups, where this is the usual notion of "finitely presented group" that is widely familiar to mathematicians.

Todd Trimble (Dec 28 2021 at 00:03):

So, on the theory or syntax side, we have small finitely complete categories $T$, finitely continuous functors between them (which could be called "interpretations" of theories), and natural transformations between those. We get a 2-category called $\mathrm{LEX}$. On the model or semantic side, we have LFP categories (definition recalled below), where the morphisms between them are right adjoints that preserve filtered colimits, and we have natural transformations between those. We get a 2-category $\mathrm{LFP}$.

Todd Trimble (Dec 28 2021 at 00:03):

Gabriel-Ulmer duality states that there is an equivalence of 2-categories

$\mathrm{LEX}^{op} \simeq \mathrm{LFP}$

where going from left to right, you take $T$ to $LEX(T, \mathsf{Set})$, and going right to left, you take $C$ to $LFP(C, \mathsf{Set})$ (this latter gives a 2-functor $\mathrm{LFP}^{op} \to \mathrm{LEX}$, but then you can apply $(-)^{op}$ to that -- here $(-)^{op}$ is the operation on 2-categories that reverses the direction on 1-cells). Thus, we obtain here a perfect duality between syntax and semantics.

Todd Trimble (Dec 28 2021 at 00:04):

Summarizing: an LFP category is a locally small and small-cocomplete category that has a small set of finitely presentable objects, such that every object is (in a canonical way) a filtered colimit of finitely presented objects. (For example, any group is a filtered colimit of finitely presented groups.)

Todd Trimble (Dec 28 2021 at 00:04):

The wonderful Gabriel-Ulmer duality can be generalized, where instead of being constrained to limits of diagrams of finite size, one considers theories involving limit diagrams of infinite size (but still bounded in size by a cardinal). The theory of locally presentable categories explores further such dualities, but now reaching into the infinitary realm.

Todd Trimble (Dec 28 2021 at 00:04):

Some of this stuff is strikingly powerful. One of my own favorite applications is to coalgebras and cocommutative coalgebras. Using the fundamental theorem of coalgebras (they are all unions of their finite-dimensional subcoalgebras), and taking advantage of the theory of LFP categories, one can show that cocommutative coalgebras are cartesian closed, develop the theory of measure coalgebras, and many other things. And, they cover a wide swath. For example, presheaf categories are LFP. Perhaps I can leave it as an exercise to show how.

Leopold Schlicht (Dec 28 2021 at 16:47):

@Todd Trimble Thank you for that amazing explanation!

You describe two constructions from LFPs two LEXs: Firstly, given an LFP $C$, assign to it the opposite of the full subcategory of $C$ consisting of finitely presentable objects. Secondly, assign to $C$ the category $\mathrm{LFP}(C, \mathbf{Set})$. Are these constructions isomorphic?

Todd Trimble said:

Summarizing: an LFP category is a locally small and small-cocomplete category that has a small set of finitely presentable objects, such that every object is (in a canonical way) a filtered colimit of finitely presented objects. (For example, any group is a filtered colimit of finitely presented groups.)

Actually, that isn't a summary, because you didn't mention these facts above. :grinning_face_with_smiling_eyes:

Which doctrine corresponds to accessible categories?

Leopold Schlicht (Dec 28 2021 at 16:53):

Why are LFPs called "presentable"? Because there's a finite-limits theory that "presents" it?

Leopold Schlicht (Dec 28 2021 at 16:56):

Or because of the "finitely presented objects"? There seems to be several ways of interpreting "presented" - probably the group-theoretic interpretation came first.

Todd Trimble (Dec 28 2021 at 17:31):

Yes, there are some things I glossed over. $\mathrm{LFP}(C, \mathsf{Set})$ means the collection of right adjoints $C \to \mathsf{Set}$ that preserve filtered colimits, and the first point is that these are the same as representables $C(c, -): C \to \mathsf{Set}$ that preserve filtered colimits. In other words, the first point is that right adjoints $C \to \mathsf{Set}$ are the same as representable functor $C(c, -): C \to \mathsf{Set}$, at least under reasonable conditions on $C$ (it's more than enough that $C$ be cocomplete).

Todd Trimble (Dec 28 2021 at 17:31):

In one direction, if $G: C \to \mathsf{Set}$ is a right adjoint, say with left adjoint $F$, then I claim $G \cong C(F1, -)$. This is because

$G \cong 1_\mathsf{Set} \circ G \cong \mathsf{Set}(1, -) \circ G= \mathsf{Set}(1, G-) \cong C(F1, -)$

where the last isomorphism comes from $F \dashv G$.

Todd Trimble (Dec 28 2021 at 17:31):

In the other direction, a representable functor $C(c, -): C \to \mathsf{Set}$ will have a left adjoint provided that $C$ admits tensoring with sets. (This means by definition that for any object $c$ of $C$ and any set $S$, there is coproduct in $C$ of an $S$-indexed set of copies of $c$. This $\coprod_S c$ is often denoted $S \cdot c$.) To see this other direction, note that

$\mathsf{Set}(S, C(c, d)) \cong \prod_S C(c, d) \cong C(\coprod_S c, d) = C(S \cdot c, d)$

and this says precisely that $- \cdot c$ is left adjoint to $C(c, -)$, so that $C(c, -)$ is a right adjoint.

Todd Trimble (Dec 28 2021 at 17:32):

The second point is that the category of filtered-colimit preserving representables $C(c, -)$ and natural transformations between them is equivalent to the opposite of the category of finitely presentable objects $c$ and morphisms between them, by the Yoneda lemma.

I think that answers your first question.

Todd Trimble (Dec 28 2021 at 17:32):

"Finitely accessible" is the condition we get from the definition of LFP category (given under "Summarizing") simply by dropping the cocompleteness condition. A good go-to example is the category of fields. I think here the finitely presentable objects are fields of finite transcendence degree over a prime field (a prime field being either $\mathbb{Q}$ or $\mathbb{Z}/(p)$. Meanwhile, however, the category of fields is not cocomplete (e.g., it lacks coproducts).

Todd Trimble (Dec 28 2021 at 17:32):

As for "accessible" itself, or $\kappa$-accessible -- that would similarly refer to one of these generalizations of locally $\kappa$-presentable I categories I was talking about, except that we again drop a cocompleteness condition. For that, let me just refer you to the nLab.

Todd Trimble (Dec 28 2021 at 17:32):

The bible of the subject, as may have already been mentioned, is the text by Adamek and Rosicky (leaving off diacritical marks), Locally Presentable and Accessible Categories. IMO, it belongs on the shelf of every category theorist.

Todd Trimble (Dec 28 2021 at 17:33):

We say "locally finitely presentable category", instead of "finitely presentable category" as Jacob Lurie might, because we're not presenting the category itself by finite graphs and congruence relations. Rather, the "locally" refers to the fact that one is talking about presentations of objects in the category. To answer your question, I think it's more because of "finitely presented objects".

Leopold Schlicht (Dec 28 2021 at 19:50):

Thank you so much!

Leopold Schlicht (Dec 28 2021 at 19:55):

Todd Trimble said:

We say "locally finitely presentable category", instead of "finitely presentable category" as Jacob Lurie might, because we're not presenting the category itself by finite graphs and congruence relations.

Strangely enough, the nLab page you link to claims the opposite:

This says equivalently that a locally presentable category $\mathcal C$ is a reflective localization $\mathcal C\hookrightarrow \mathrm{PSh}(S)$ of a category of presheaves over $S$. Since here $\mathrm{PSh}(S)$ is the free colimit completion of $S$ and the localization imposes relations, this is a presentation of $\mathcal C$ by generators and relations, hence the name (locally) presentable category.

Todd Trimble (Dec 28 2021 at 19:56):

Let me get back to you on this.

Todd Trimble (Dec 28 2021 at 19:58):

So my understanding is consonant with Remark 1.17 in the nLab here.

Todd Trimble (Dec 28 2021 at 20:18):

I guess I'll also adduce Remark 2.3 here.

I think there are probably public discussions, maybe on the old Categories list, that discuss/debate this terminology, but I can't quite put my finger on them at the moment. Of course, bear in mind that the nLab is edited by many people (or many states of the same person), and there's no organizing committee at the head that makes sure that it's internally consistent everywhere.

Mike Shulman (Dec 28 2021 at 21:47):

The historically original terminology, and the only one in use for many decades, was "locally presentable category", with the motivation that Todd mentioned. Then some people were apparently too lazy to write "locally" all the time, so they started saying just "presentable category".

One can try to justify this a posteriori by claiming that the description of a locally presentable category as a reflective subcategory of a presheaf category is a kind of "presentation" of it, but in my opinion this is execrable. A presentation of a category should mean a coequalizer of maps between free categories in $\rm Cat$, just as a presentation of a group is a coequalizer of maps between free groups in $\rm Grp$. One could argue for replacing "coequalizer" with something more 2-categorical like "coinverter", and a reflective subcategory is indeed a coinverter, but not in the 2-category $\rm Cat$, only in the 2-category of cocomplete categories and left adjoints. So at the most we can regard this as a presentation of $C$ as a cocomplete category, so if that is the motivation one should say "presentable cocomplete category".

Nathanael Arkor (Dec 28 2021 at 22:35):

I find "presentable category" unideal for the reasons stated above, but "locally presentable category" is also unfortunate, because "locally" is typically used to mean "with respect to homs" (e.g. locally complete), and here it means something entirely different.

Zhen Lin Low (Dec 28 2021 at 22:39):

Actually, I thought other than "locally small", "locally X" often commonly means all slices are X, c.f. "locally cartesian". Incidentally, there is a scenario where we want all slices to be essentially small...

Nathanael Arkor (Dec 28 2021 at 22:44):

That's a good point. "Locally" in the sense of hom-structure is often used for 2-categorical structure, whereas "locally" in the sense of slices is often used for finitely complete structure.

Nathanael Arkor (Dec 28 2021 at 22:45):

We're a little stuck with both of them, unfortunately. But it would be nice if there was some alternative terminology to disambiguate the two.

Todd Trimble (Dec 28 2021 at 22:56):

The explanation that Leopold pointed to reads to me like after-the-fact folk etymology. Aside from Mike's objection (which I pretty much agree with), I find it objectionable because it's confusing, being at odds with explanations elsewhere in the nLab.

Zhen Lin Low (Dec 28 2021 at 22:59):

I think "locally" for slicewise is apt. I am under the impression that a category is the pseudocolimit of its slices (with the pushforward action), so this fits with the topological use of "locally". Similarly, in my view, "locally κ-presentable category" refers to the fact that the objects in the category are colimits of κ-presentable objects (so it is the objects that are locally presentable!), and is in the tradition of saying "X category" for "categories of X objects".

Nathanael Arkor (Dec 29 2021 at 01:31):

I would expect "locally presentable" to mean "each slice is presentable" in that case.

Nathanael Arkor (Dec 29 2021 at 01:32):

While they may both capture similar intuitions, they seem different enough to me that I think they ought to use separate terminology.

Nathanael Arkor (Dec 29 2021 at 01:32):

Though in practice, I don't think it causes confusion.

Mike Shulman (Dec 29 2021 at 01:32):

Nathanael Arkor said:

We're a little stuck with both of them, unfortunately. But it would be nice if there was some alternative terminology to disambiguate the two.

Fortunately, it's rare to find values of X for which "locally X" is interesting for both meanings of "locally". Or, at least, for any given value of X, one of the two meanings of "locally X" is so much more common that everyone knows what it means. If necessary one can use words like "homwise" and "slicewise".

Mike Shulman (Dec 29 2021 at 01:34):

Historically, I think "locally" was sometimes used with a third meaning that was something like "for a generating set of objects". For instance, in some very old papers one can find "locally small" used to mean something roughly like "has a small generating set". I think this must be the origin of "locally presentable" = "having a generating set of presentable objects".

Mike Shulman (Dec 29 2021 at 01:35):

So I suppose a more modern and explicit terminology would be something like "presentably generated category", but I would hesitate to introduce yet another name for them.

Leopold Schlicht (Jan 17 2022 at 15:50):

Question 1: Can somebody provide a mathematical reference for the following philosopical thoughts of Lawvere (found in his Adjointness in Foundations):
lawvere.png
This could be interesting, but I don't know what he is talking about, since he doesn't give precise definitions.

Question 2: Gabriel-Ulmer duality is very similar to Makkai duality: both tell us how to reconstruct a theory from its category of models. Furthermore, the paper On the duality between varieties and algebraic theories by Adámek, Lawvere, and Rosický proves that the "models" functor from the opposite of the 2-category of Cauchy complete algebraic theories to the 2-category of varieties is a biequivalence. Are there any other results of that type? I guess so because Todd Trimble wrote:

The wonderful Gabriel-Ulmer duality can be generalized, where instead of being constrained to limits of diagrams of finite size, one considers theories involving limit diagrams of infinite size (but still bounded in size by a cardinal). The theory of locally presentable categories explores further such dualities, but now reaching into the infinitary realm.

Is there somewhere an overview of all these results?

Nathanael Arkor (Jan 17 2022 at 17:36):

Regarding the last question, these results are subsumed by Centazzo–Vitale's A duality relative to a limit doctrine and, more recently, the paper Accessibility and presentability in 2-categories by @Ivan Di Liberti and @fosco.

Leopold Schlicht (Jan 17 2022 at 18:59):

Thanks, but are you sure they subsume Makkai duality? Anyway, I'd be more interested in a comprehensive list of concrete dualities than a big general theorem.

Nathanael Arkor (Jan 17 2022 at 20:50):

Sorry, I meant to say that it subsumes Gabriel–Ulmer and Adámek–Lawvere–Rosický duality. I'm not sure whether Makkai duality fits into a more general framework that is in the literature.