Contents

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.

This is (AdamekRosicky, theorem 6.28).

## 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.

• Lie integration, which constructs a simply connected Lie group from a finite-dimensional real Lie algebra. The coreflector is Lie differentiation (taking a Lie group to its associated Lie algebra), and the counit is the natural map to a given Lie group $G$ from the universal covering space of the connected component at the identity of $G$.

• In a recollement situation, we have several reflectors and coreflectors. We have a reflective and coreflective subcategory $i_*: A' \hookrightarrow A$ with reflector $i^*$ and coreflector $i^!$. The functor $j^*$ is both a reflector for the reflective subcategory $j_*: A'' \hookrightarrow A$, and a coreflector for the coreflective subcategory $j_!: A'' \hookrightarrow A$.