nLab PrCat

Idea

When working with locally presentable categories, one is typically interested only in the colimit-preserving functors between them, hence (by the adjoint functor theorem) equivalently the left adjoint functors.

One hence considers the very large category $PrCat$ whose objects are locally presentable categories, and whose morphisms are left adjoint functors.

The analog of this

• in (∞,1)-category theory is Pr(∞,1)Cat;

• in model category-theory is Ho(CombModCat).

Related concepts

• $Cat_{Adj}$

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.

$\phantom{A}$(n,r)-categories$\phantom{A}$	$\phantom{A}$toposes$\phantom{A}$	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	Gabriel–Ulmer’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)-category theory	(∞,1)-toposes	locally presentable (∞,1)-categories		Simpson’s theorem	(∞,1)-presheaf (∞,1)-categories	accessible (∞,1)-categories