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I'm trying to understand the notion of an accessible category. This isn't the first time I've tried to do this; but every time I try to make sense of the definition, I become perturbed by the following issue: not every small category is accessible.

You can find a definition of accessible category at nLab. In brief a (possibly large) category C is accessible if there's a regular cardinal $\kappa$ such that C has $\kappa$-directed colimits, and that there's a set of $\kappa$-compact objects so that every object C is a $\kappa$-directed colimit of things in this set. Thus, the behavior of the category C is somehow "controlled" by a small subcategory; roughly speaking, all objects of C "look like" filtered colimits of objects in the small subcategory.

Any small category is of course "controlled" by a small subcategory, namely itself, so you'd think that all small categories are accessible. Not quite! The correct statement is:

A small category is accessible if and only if it is idempotent complete

This is proved (I think) in Adamek & Rosicki, Locally Presentable and Accessible Categories. There is a also a proof of an $\infty$-category version of this in Lurie's Higher Topos Theory.

My question is not about the proof of this claim (which I think I understand), but about the underlying motivation for the notion of accessible category. Basically, I'd like one of two things:

  1. Make me understand why it's such a good thing that not every small category is accessible, or

  2. Tell me that "accessible category" is not exactly the right idea, and that there's a generalization of it which includes all small categories a special case.

(Note: the class of accessible categories is closed under a bunch of constructions, such as taking undercategories, or taking functors from a fixed small category. The generalization of 2 ought to have the same properties.)

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5 Answers 5

up vote 11 down vote accepted

To me, the "obvious" guess at (2) would be a category whose idempotent-splitting-completion (aka "Cauchy completion" or "Karoubi envelope") is accessible. While I don't have an explicit counterexample, I doubt that these have all the same good properties. The two properties you mention are special cases of closure under pseudo-limits, but the pseudo-limit of a Karoubi envelope is not in general the same as the Karoubi envelope of the pseudo-limit. For instance, let $C$ be the "walking split idempotent", containing two objects $x$ and $y$ with $y$ a retract of $x$, let $F,G\colon C\to Set$ send $x$ to a set $S$ and $y$ to a nonempty subset $T\subseteq S$ with a chosen retraction $S\to T$, and let $\alpha,\beta\colon F\to G$ be natural transformations which are equal on $S$ but not on all of $T$. Then the equifier of $\alpha$ and $\beta$ consists only of $y$, whereas if $C'\subseteq C$ contains only $x$, then the equifier of $\alpha$ and $\beta$ restricted to $C'$ is empty.

One answer to (1) is to consider some of the other characterizations of accessible categories. For instance, a category is accessible iff:

  • it is the category of $\kappa$-flat functors from some small category to Set (for some $\kappa$), or iff
  • it is the category of models in Set of a small sketch, or iff
  • it is the category of models in Set of some suitable logical theory.

These sorts of categories clearly always have split idempotents.

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The alternate characterizations are a pretty compelling answer to (1). I had sort of seen the first characterization on your list in Makkai & Pare, but I guess it didn't really sink in. –  Charles Rezk Jan 10 '10 at 15:13
    
Mike, surely the first of your three bullet points ("the category of flat functors from small category to Set") is a characterization of finitely accessible categories. (I spotted this because I made exactly the same slip myself a month or two ago, on another site where you and I were discussing accessible categories.) –  Tom Leinster Jan 10 '10 at 16:58
    
Thanks Tom, fixed. –  Mike Shulman Jan 11 '10 at 4:45

These are just some half-baked ideas, which could be totally off track, and I am in a bit of a hurry. But anyways, food for thought.

For such questions it's very important to keep track at the same time of what the morphisms (and natural transformations) are. As Charles says in a comment, the standard morphisms between accessible categories are the accessible functors. I assume that accessible functors between small accessible categories are all functors, since the small accessible categories are the idempotent complete categories and all functors preserve retracts. I also believe that functors from C to D, where D is idempotent complete, are the same as functors from the idempotent completion of C to D.

If that is all true, then we could just as well say that indeed, all small categories are accessible–provided that we redefine the notion of an accessible functor from C to D as an accessible functor in the ordinary sense between their idempotent completions. This sounds artificial, but the ordinary definition of accessible functor doesn't seem very good when the domain fails to have κ-filtered colimits for every κ, so maybe there is another point of view on this definition which generalizes to the definition in terms of idempotent completions. For instance, is a functor between the idempotent completions of C and D (small categories) the same as a functor from presheaves on D to presheaves on C which has adjoints on both sides?

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An essential property of accessible categories is that if $C$ is $\kappa$-accessible then it is also $\lambda$-accessible for many $\lambda > \kappa$ in a way that is essentially independent of $C$. (The characterization of such pairs $\kappa,\lambda$ is complicated, see Adamek & Rosicky for details.) This is why the particular cardinal $\kappa$ which verifies accessibility is in many ways irrelevant. This property fails badly without the existence of appropriate directed colimits.

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This isn't really a fulll answer, but there is a obvious guess for (2). Consider those categories for which there exists a cardinal $\kappa$ and a set of $\kappa$-compact objects such that every object is a $\kappa$-directed colimit of things in this set. You just drop the requirement that every $\kappa$-directed colimit exists. This includes, in particular, all small categories.

It seems like this class still has the properties that you mentioned. Are there any other nice properties that accessible categories have that this class doesn't?

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Well, there are other properties. I neglected to mention that one of the main reasons for considering accessible categories is to talk about accessible functors between such. And accessible functor is defined as one which commutes with kappa-directed colimits for some big kappa. I don't know what happens if you don't require that the colimits exist. –  Charles Rezk Jan 9 '10 at 21:16
    
Everything works out fine if you don't require that the colimits exist (cf. Cisinski's Les Prefaisceaux comme modeles des types d'homotopie). –  Harry Gindi Jul 14 '10 at 6:52

In a accessible category idempotents split (the usual coker $Cok(1, f)$ for split a idempotent $f$ is a colimit of a filtrant diagram). And any small category with split idempotents is accessible.

See: p. 71 (point 2.4, 2.6) on

J. Adamek and J. Rosicky, Locally Presentable and Accessible Categories Cambridge: Cambridge University Press, 1994

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