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quickly added at accessible category parts of the MO discussion here. Since Mike participated there, I am hoping he could add more, if necessary.
Since Mike participated there, I am hoping he could add more, if necessary.
Thanks to Mike for editing the entry now, nicely!
I have touched accessible category. Provided numbered environments, collected the references and harmonized the pointers to them.
That reminds me I never did anything about Mike’s recommendation here. Could those four bullet points at sketch be transferred to accessible category?
Certainly. What would you be worried about? Small parts of this is already at the entry.
But the text needs more links, notably for multireflective subcategory and weakly reflective subcategory.
I added in a link to the Adamek, Borceux, Lack, Rosicky paper on a classicfation of accessible categories. (I duplicated the link at ind-object.
Apologies if the nForum is not the place to debate history, but I believe that the terms “accessible functor” and “accessible category” were used long before Makkai and Paré: in SGA 4, Exposé I no. 9.2-4. Grothendieck defines these ’after a suggestion by Deligne’. Grothendieck actually develops most of the theory of accessible categories right then and there- including giving the standard equivalent characterizations (a special case of 9.18) and proving that accessible categories are closed under certain homotopy limits and right lax limits (Theorem 9.22, albeit using the language of fibered categories not (lax) limits). He also proves various (ind/pro-)representability and (ind/pro-)adjoint functor theorems in the previous sections, some combinations of which include the familiar ones about presentable categories (though it isn’t spelled out in this way).
I’m very confused by the complete lack of citations to SGA4 in Makkai-Pare and Adameck-Rosicky, both of which came way after. There’s even some argument to be made about that Gabriel-Ulmer should have given Grothendieck a nod for their work on presentable categories (but, though the lectures for SGA4 took place in the early 60s, the Springer book was after Gabriel-Ulmer’s, so that’s fair.) Many places (not just the nLab) seem to ignore this Exposé altogether in the history of presentable/accessible categories, and that makes no sense to me…
Apologies for the rant-ish nature of this post- I’m not actually upset, just perplexed. I’m wondering if I’m just out of the loop on some aspect of this history, and someone can help me set it straight.
@Dylan If I remember correctly, SGA4 is not written entirely by Grothendieck but is rather a compilation of lecture notes in a seminar convened by Grothendieck. Certain parts of it have been attributed to different authors (in particular, I remember reading that Artin authored a large part of SGA4, for example, including a pretty difficult proof of a desingularization theorem in commutative algebra).
It’s possible that Makkai or Pare participated in those seminars.
However, I did take a look at the introduction to Makkai-Pare, and it does indeed cite SGA4 quite extensively in its introduction (see page 6 of the print edition), but it makes no reference to the specific person who presented the idea to the seminar.
I’ll quote it here for you:
Another important source for the present work is [SGA4], especially Section 9 of Expose I. In Sections 2.3 and 5.2 below, we will state the connections of the present work and [SGA4] in detail; here we mention the main points only.
The expression “accessible category” is an adaptation of related terminology introduced in [SGA4], although our notion of accessible category is not named, in this or any other way, in [SGA4]. The notion “$\pi$-accessible category” is not equivalent to our “$\pi^+$-accessible category”, it is strictly weaker than ours. On the other hand, the main result of the relevant part of [SGA4], Theorem 9.22 of Expose I, contains, in the form of a rather technical condition, an assumption of accessibility in our sense. This theorem is an important forerunner (in fact, an incomplete statement) of one of our main results, the Limit theorem (5.1.6).
The crucial notion of accessible functor is the same as the one so named in [SGA4]: a functor (between accessible categories) is accessible if, for some cardinal $\kappa$, it preserves $\lt \kappa$-filtered colimits. This notion is related to the notion of a functor with a rank; see [B] and [G/U]. Note that, for the purposes of Lair’s theorem, one does not need a notion of accessibility of functors, since the theorem talks about a single category at a time. In this paper however, our interest is in the connections between, and constructions upon, various accessible categories simultaneously; in [SGA4], we find a similar concern. Therefore, we are interested in the right category, in fact 2-category, of accessible categories. The 2-category $\mathcal{Acc}$ of accessible categories is introduced in the present work for the first time.
Apologies if the nForum is not the place to debate history,
It’s not frowned upon, but you are encouraged to add useful comments like the ones you make to the $n$Lab entry itself. The idea is that this way many more people will eventually see and appreciate it.
(The $n$Forum is to the $n$Lab like the “talk” pages for Wikipedia. Eventually the goal of any discussion here and there ought to be to improve the relevant entry.)
@Urs I just included the part from Makkai-Pare (I edited my post above). The original idea comes from SGA4, but Makkai and Pare give the modern definition.
Thanks! Why don’t you paste this into the entry, maybe in a “History” section?
@Urs Eh, I don’t think that whole thing belongs in the article. Probably if anyone cares, one would put something like “The idea of accessibility was first discussed in SGA4 Expose I, section 9, but the modern definition comes from Makkai-Pare blah blah blah”.
If Dylan wants to add it or you do, be my guest. It doesn’t really bother me either way. The term accessible category is not due to Makkai-Pare, but the modern definition of it is due to them.
@Harry Excellent! I couldn’t get a hold of Makkai-Pare and made the silly mistake of trusting the Math Reviews list of references and Adameck-Rosicky’s introduction. Glad to know they address this! I will have to think a bit to see just how the definitions differ (I guess the way in which \pi-accessible objects ’generate’ is maybe different), but that’s okay.
(As for attribution: SGA4 gives attribution next to each Expose. Grothendieck and Verdier (guess I forgot to mention Verdier) are listed next to Expose I. Artin’s name appears next to Expose’s IX-XVI, as you say.)
@Urs Good to know- I didn’t want to edit the page because I was worried I was missing something. Turns out I was!
Thanks again!
Oy vey- I was more tired than I thought! I forgot about “Artin” when searchning for SGA4 in references… and missed it in both Adameck-Rosicky and Makkai-Pare. So this was all a dumb mistake on my part, and I apologize for the wasted time… Adameck-Rosicky also make some historical remarks after introducing accessible categories. They give one sentence to SGA4, then say the definition was originally due Lair and independently Rosicky. Seems like this was a popular thing to define.
Maybe we should just forget this whole thing… Sorry for the unjustified fuss!
Adamek (in fact Adámek), not Adameck.
Dylan, all the more, it shows that it would it be useful to add some such discussion of the history of the idea to the entry.
Are all accessible categories well-powered? I didn’t see any remarks about this one way or the other on a quick skim through Adamek-Rosicky, but I could have missed it.
It seems you already clarified this in the entry well-powered category
No, I didn’t write anything about accessible categories being well-powered, only well-copowered (and I wrote the same thing at accessible category).
Yes, that was a typo, thanks. It was correct at accessible category; somehow I guess I fixed it in one place but not the other.
Hmm, this paper claims that all accessible categories are well-powered and that it’s in Makkai-Paré. I guess I need to go back and look there again.
Makkai–Paré Prop 6.1.3 is the claim that if an accessible category has pushouts then it is co-well-powered. There is no reference to “well-powered” in Makkai–Paré, but perhaps the result is not referred to in such terms. Searching for “subobject” doesn’t return anything useful either. Perhaps Barr got that from somewhere else?
I couldn’t find it in Makkai-Paré either. Time to ask MO.
Thanks for catching that! Seems to be a false statement starting out at theory of presheaf type ending here by copy-and-paste. I replaced it by other statements taken over from theory of presheaf type and sketch.
Enriched version
added publication data to
and pointer to
added (here) statement that AccCat has all PIE 2-limits, following discussion in another thread (here)
For the enriched case I added pointer to (Thm. 5.5 in)
Thanks. I made that read “2-pullbacks along isofibrations”.
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