It is a means to handle -categories that are not essentially small in terms of small data.
An accessible -category is one which may be large, but can entirely be accessed as an -category of “conglomerates of objects” in a small -category – precisely: that it is a category of -small ind-objects in some small -category .
Let be a regular cardinal. spring
A (∞,1)-category is -accessible if it satisfies the following equivalent conditions:
of with the (∞,1)-category of ind-objects, relative , in .
The notion of accessibility is mostly interesting for large (∞,1)-categories. For
Generally, is called an accessible -category if it is -accessible for some regular cardinal .
These conditions are indeed equivalent.
This is HTT, def. 188.8.131.52.
If is an accessible -category then so are
This is HTT section 5.4.4, 5.4.5 and 5.4.6.
This is HTT, section 5.4.6.
The -category has all small (∞,1)-limits and the inclusion
This is HTT, proposition 184.108.40.206.
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 theory of accessible 1-categories is described in
The theory of accessible -categories is the topic of section 5.4 of