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).
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.
Created on July 6, 2018 at 15:49:04. See the history of this page for a list of all contributions to it.