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.

]]>Maybe the intended statement was that of Proposition 4.1 at theory of presheaf type? (Which characterizes finitely accessible categories, not theories of presheaf type.) But I was wondering if it might have been something else. ]]>

Fixed statement of adjoint functor theorem, and removed query box.

]]>Added John Bourke’s recent arXiv paper.

By the way, somebody should attend to Emily’s query concerning the adjoint functor thm as stated in the entry.

]]>Added some citations to recent papers about abstract elementary classes, many of which contain useful facts about general accessible categories.

]]>Added well-poweredness.

]]>I couldn’t find it in Makkai-Paré either. Time to ask MO.

]]>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?

]]>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.

]]>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.

]]>No, I didn’t write anything about accessible categories being well-powered, only well-copowered (and I wrote the same thing at accessible category).

]]>It seems you already clarified this in the entry well-powered category

]]>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.

]]>Added remarks about well-copoweredness

]]>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.

]]>Adamek (in fact Adámek), not Adameck.

]]>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!

]]>@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!

]]>@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.

]]>Thanks! Why don’t you paste this into the entry, maybe in a “History” section?

]]>@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.

]]>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.)

]]>@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.