nLab PrCat

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Applications

Compact objects

objects $d \in C$ such that $C(d,-)$ commutes with certain

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

The analog of this

Locally presentable categories: possibly- generated under by under . Equivalently, of . Accessible categories omit the cocompleteness requirement; toposes add the requirement of a localization.

$\phantom{A}$$\phantom{A}$$\phantom{A}$$\phantom{A}$locally presentableloc finitely preslocalization theoremaccessible
Porst’s theorem