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.”
See also at categorical model theory.
See at Functor category – Accessibility.
Locally presentable categories: Cocomplete possibly-large categories generated under filtered colimits by small generators under small relations. Equivalently, accessible localizations of free cocompletions. Accessible categories omit the cocompleteness requirement; toposes add the requirement of a left exact localization.
|(n,r)-categories||toposes||locally presentable||loc finitely pres||localization theorem||free cocompletion||accessible|
|(0,1)-category theory||locales||suplattice||algebraic lattices||Porst’s theorem||powerset||poset|
|category theory||toposes||locally presentable categories||locally finitely presentable categories||Adámek-Rosický’s theorem||presheaf category||accessible categories|
|model category theory||model toposes||combinatorial model categories||Dugger’s theorem||global model structures on simplicial presheaves||n/a|
|(∞,1)-topos theory||(∞,1)-toposes||locally presentable (∞,1)-categories||Simpson’s theorem||(∞,1)-presheaf (∞,1)-categories||accessible (∞,1)-categories|
The term accessible category is due to
The standard textbook on the theory of accessible categories is
which further stratifies the accessible categories in terms of sound doctrines.
Accessible categories in the context of categorical model theory are further discussed in