objects such that commutes with certain colimits
the category has -directed colimits (or, equivalently, -filtered colimits), and
If satisfies these properties for some , we say that it is -accessible.
Unlike for locally presentable categories, it does not follow that if is -accessible and then is also -accessible. It is true, however, that for any accessible category, there are arbitrarily large cardinals such that is -accessible.
Equivalent characterizations include that is accessible iff:
it is of the form for small, i.e. the -ind-completion of a small category, for some .
it is of the form for small and some , i.e. the category of -flat functors from some small category to .
it is the category of models (in ) of a suitable type of logical theory.
The relevant notion of functor between accessible categories is
(preservation of accessibility under inverse images)
This appears as HTT, corollary A.2.6.5.
(accessibility of fibrations and weak equivalences in a combinatorial model category)
This appears as HTT, corollary A.2.6.6.
(closure under limits)
Every accessible functor satisfies the solution set condition, and every left or right adjoint between accessible categories is accessible. Therefore, the adjoint functor theorem takes an especially pleasing form for accessible categories: a functor is a left (resp. right) adjoint iff it is accessible and preserves all small colimits (resp. limits).
Makkai-Paré say that this means accessibility is an “almost pure smallness condition.”
|(n,r)-categories…||satisfying Giraud's axioms||inclusion of left exaxt localizations||generated under colimits from small objects||localization of free cocompletion||generated under filtered colimits from small objects|
|(0,1)-category theory||(0,1)-toposes||algebraic lattices||Porst’s theorem||subobject lattices in accessible reflective subcategories of presheaf categories|
|category theory||toposes||locally presentable categories||Adámek-Rosický’s theorem||accessible reflective subcategories of presheaf categories||accessible categories|
|model category theory||model toposes||combinatorial model categories||Dugger’s theorem||left Bousfield localization of global model structures on simplicial presheaves|
|(∞,1)-topos theory||(∞,1)-toposes||locally presentable (∞,1)-categories|| |
|accessible reflective sub-(∞,1)-categories of (∞,1)-presheaf (∞,1)-categories||accessible (∞,1)-categories|
The term accessible category is due to
The standard textbook on the theory of accessible categories is