nLab
coreflective subcategory

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Context

Category theory

category theory

Concepts

Universal constructions

Theorems

Extensions

Applications

Notions of subcategory

Modalities, Closure and Reflection

modal type theory, modal logic

closure operator, universal closure operator

Examples

Contents

Definition

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

C←R↪iD.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)
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