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:
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Make me understand why it's such a good thing that not every small category is accessible, or
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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.)
Best Answer
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:
These sorts of categories clearly always have split idempotents.