nLab
coreflective subcategory

Context

Category theory

Concepts

Universal constructions

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Theorems

Extensions

Applications

Notions of subcategory

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Modalities, Closure and Reflection

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Examples

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Contents

Definition

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

CRiD. 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 CC a locally presentable category, every full subcategory DCD \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 GG from the universal covering space of the connected component at the identity of GG.

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

References

Last revised on July 11, 2018 at 11:33:41. See the history of this page for a list of all contributions to it.