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‘An accessible category is a possibly large category which is however essentially determined by a small category.’ Except that it’s a stricter notion than that! After all, a small category is, obviously, essentially determined by itself, but a small category is accessible only if it is idempotent-complete.
Is there a notion like accessible category that captures this idea but includes all small categories? Ideally, I would like a notion of ‘good’ category such that:
(These are in order of importance; bonus points if the first four properties hold in predicative mathematics but the last property is impredicative.)
Failing that, is there a good reason why only idempotent-complete categories should be considered as ‘essentially determined by a small category’?
Good question. There was some discussion of this initiated by, I think, Charles rezk on MO at some point. But I don’t have the link right now.
Thanks, Urs, I found the MO discussion. It doesn’t really give me an answer that I like, but Mike Shulman explains why one obvious guess is not very nice. (That lowers the odds that Mike will have a nice answer for me now.)
It does, doesn’t it! (-: I think the way I resolve this issue in my head is that I don’t agree with the statement you quoted (’an accessible category is a possibly large category which is however essentially determined by a small category’), at least not as a definition of ’accessible category.’
I think pretty much any category that arises in mathematics is ’essentially determined by a small category’ in some sense. For instance, if the objects in your category are models of some (small) first-order theory, then the category is ’essentially determined’ in some sense by the walking model of that theory, which is a small category. Accessible categories are ones that are ’essentially determined by a small category’ in a particular, very precise, way.
OK, so you agree with me that it is a stricter notion than is implied by my opening quotation (which I took straight from the nLab article, in case that wasn’t clear).
I think pretty much any category that arises in mathematics is ’essentially determined by a small category’ in some sense.
I agree with that, but it would be nice to have a definition (and maybe a metatheorem about what can actually arise in a formal development?) that makes this precise.
Without that, at least I now better understand why people focus on accessible categories.
Yes, I think that sentence should be changed on the nLab.
Here’s one natural definition of large categories determined by a small amount of data. Pick a small quiver C; it determines a logical signature whose types are its objects and whose operations are its morphisms. Now write down a small number of axioms in that signature. This determines a full subcategory of [C,Set] whose objects are those satisfying the axioms. Lots of large categories are of this form; the accessible ones are just those where the axioms are restricted to have a certain specific form (“limit+colimit theories,” aka sketches). For additional generality, you can restrict to the core of such a category and then add new morphisms in some way, i.e. consider the “exact completion” in a 2-categorical sense.
In material set theory, you can say something easier and more trivial, since there a large category is just a couple of proper classes, and in ZFC every proper class is determined by a small amount of data. But maybe that’s further removed from what you were getting at.
in ZFC every proper class is determined by a small amount of data
Actually, I’m not sure what you mean by this. As a metatheorem, it’s true that every sentence in the first-order language of ZFC with one free variable is determined by a finite amount of data, but that’s not what I’m getting at, and it’s much stronger than what you said, so it can’t be what you’re getting at either. But I don’t see any other sense in which every proper class is determined by a small amount of data.
That is basically what I meant; the data just goes up from “finite” to “small” if you allow additional parameters in the sentence. Which you have to do, if you want to consider, for example, arbitrary slice categories of large categories. If that isn’t what you’re getting at, can you say a bit more about what you are looking for?
There are still only finitely many parameters, so that’s still only a finite amount of data.
But that is, it seems to me, entirely irrelevant. Anything that can actually be written down in a formal system (at least the recursively enumerable formal systems that we actually use as foundations) is given by a finite amount of data. But I don’t think that this is at all the intuition behind the idea that an accessible category is determined by a small amount of data.
When I mentioned metatheorems parenthetically, I was thinking of … something that I now realise, when I come to write this comment, was trivial. So forget about that.
That still leaves your biggest paragraph from comment #6. I’m still thinking about that.
There are still only finitely many parameters, so that’s still only a finite amount of data.
Well, given any amount of data at all, the set containing all that data is a single object, and 1 is finite; does that make any amount of data into a finite amount of data? In an ambient material set theory, I’d be inclined to only call a set “a finite amount of data” if it’s hereditarily finite.
Anything that can be completely written down in a formal system is given by a finite amount of data. But I want to talk about things like “Set/X for any set X” which is not “completely written down” since X is a parameter. I don’t regard Set/X for some arbitrary set X as given by a finite amount of data because the set X may not be finite (in any sense).
I would say that “Set/X for any set X” does not in fact give anything at all until you say what is. Unless you want it to give the map , but now I’d say that that is given by a finite amount of data.
But again, I don’t think that this is relevant to the sense in which an accessible category is given by a small category.
This is getting kind of linguistically quibbly, so we should probably stop. (-: Possibly you’re right, although I don’t want to spend the time right now to figure out exactly what the right thing to say is. If there is one.
I adjusted the intro of accessible category very slightly, to make it a bit less misleading.
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