nLab
PrCat

Contents

Contents

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 PrCatPrCat whose objects are locally presentable categories, and whose morphisms are left adjoint functors.

The analog of this

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.

A\phantom{A}(n,r)-categoriesA\phantom{A}A\phantom{A}toposesA\phantom{A}locally presentableloc finitely preslocalization theoremfree cocompletionaccessible
(0,1)-category theorylocalessuplatticealgebraic latticesPorst’s theorempowersetposet
category theorytoposeslocally presentable categorieslocally finitely presentable categoriesAdámek-Rosický‘s theorempresheaf categoryaccessible categories
model category theorymodel toposescombinatorial model categoriesDugger's theoremglobal model structures on simplicial presheavesn/a
(∞,1)-category theory(∞,1)-toposeslocally presentable (∞,1)-categoriesSimpson’s theorem(∞,1)-presheaf (∞,1)-categoriesaccessible (∞,1)-categories

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