nLab
coreflective subcategory

Definition

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

$$C\stackrel{\stackrel{i}{\hookrightarrow }}{\underset{R}{\leftarrow }}D.$$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 space?s into Top, where the right adjoint “kelleyfies”

References

• Jiri Adamek, Jiří Rosický, Locally presentable and accessible categories, London Mathematical Society Lecture Note Series 189