category theory

# Contents

## Definition

A coreflective subcategory is a full subcategory whose inclusion functor has a right adjoint $R$ (a cofree functor):

$C\stackrel{\stackrel{i}{↪}}{\underset{R}{←}}D\phantom{\rule{thinmathspace}{0ex}}.$C \stackrel{\overset{i}{\hookrightarrow}}{\underset{R}{\leftarrow}} D \,.

The dual concept is that of a reflective subcategory. See there for more details.

## Properties

###### Theorem

Vopěnka's principle is equivalent to the statement:

For $C$ a locally presentable category, every full subcategory $D↪C$ which is closed under colimits is a coreflective subcategory.

This is (AdamekRosicky, theorem 6.28).

## Examples

• the inclusion of Kelley space?s into Top, where the right adjoint “kelleyfies”

## References

• Robert El Bashir, Jiri Velebil, Simultaneously Reflective And Coreflective Subcategories of Presheaves (TAC)

Revised on December 10, 2012 11:07:45 by Urs Schreiber (89.204.138.8)