Background
Basic concepts
equivalences in/of -categories
Universal constructions
Local presentation
Theorems
Extra stuff, structure, properties
Models
The notion of accessible -category is the generalization of the notion of accessible category from category theory to (∞,1)-category theory.
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 .
A -accessible -category which in addition has all (∞,1)-colimits is called a locally ∞-presentable or a -compactly generated (∞,1)-category.
Let be a regular cardinal.
A (∞,1)-category is -accessible if it satisfies the following equivalent conditions:
There is a small (∞,1)-category and an equivalence of (∞,1)-categories
of with the (∞,1)-category of ind-objects, relative , in .
The -category
has all -filtered colimits
the full sub-(∞,1)-category of -compact objects is an essentially small (∞,1)-category;
generates under -filtered (∞,1)-colimits.
The -category
has all -filtered colimits
there is some essentially small sub-(∞,1)-category of -compact objects which generates under -filtered (∞,1)-colimits.
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.
For the first few this is HTT, prop. 5.4.2.2. The last one is in HTT, section 5.4.3.
An (∞,1)-functor between accessible -categories that preserves -filtered colimits is called an accessible (∞,1)-functor .
Write for the 2-sub-(∞,1)-category of (∞,1)Cat on
those objects that are accessible -categories;
those morphisms for which there is a such that the (∞,1)-functor is -continuous and preserves -compact objects.
So morphisms are the accessible (∞,1)-functors that also preserves compact objects. (?)
This is HTT, def. 5.4.2.16.
If is an accessible -category then so are
for a small simplicial set the (∞,1)-category of (∞,1)-functors ;
for a small diagram, the over quasi-category and under-quasi-category .
This is HTT section 5.4.4, 5.4.5 and 5.4.6.
The (∞,1)-pullback of accessible -categories in (∞,1)Cat is again accessible.
This is HTT, section 5.4.6.
Generally:
This is HTT, proposition 5.4.7.3.
Locally presentable categories: Cocomplete possibly-large categories generated under filtered colimits by small generators under small relations. Equivalently, accessible reflective localizations of free cocompletions. Accessible categories omit the cocompleteness requirement; toposes add the requirement of a left exact localization.
Theory of accessible 1-categories:
Theory of accessible -categories:
See also:
Last revised on October 1, 2021 at 04:46:44. See the history of this page for a list of all contributions to it.