category theory

# Contents

## Definition

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

$C \stackrel{\overset{i}{\hookrightarrow}}{\underset{R}{\leftarrow}} D \,.$

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 $D \hookrightarrow C$ which is closed under colimits is a coreflective subcategory.

## Examples

• the inclusion of Kelley spaces into Top, where the right adjoint “kelleyfies”

• the inclusion of torsion abelian groups into Ab, where the right adjoint takes the torsion subgroup.

• the inclusion of groups into monoids, where the right adjoint takes a monoid to its group of units.

## References

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

Revised on September 8, 2014 21:59:13 by David Corfield (91.125.67.210)