nLab
coreflective subcategory

Contents

Context

Category theory

Notions of subcategory

Modalities, Closure and Reflection

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.

Characterizations

Proposition

(equivalent characterizations)

Given any pair of adjoint functors

(LR):BRLA (L \dashv R) \;:\; B \underoverset {\underset{R}{\longrightarrow}} {\overset{L}{\longleftarrow}} {\bot} A

the following are equivalent:

  1. The left adjoint LL is fully faithful. (In this case AA is equivalent to its essential image in BB under LL, a full coreflective subcategory of BB.)

  2. The unit η:1 ARL\eta : 1_A \to R L of the adjunction is a natural isomorphism of functors.

  3. The comonad (LR,LηR,ϵ)(L R, L\eta R,\epsilon) associated with the adjunction is idempotent, the left adjoint LL is conservative, and the right adjoint RR is essentially surjective on objects.

  4. If SS is the set of morphisms ss in BB such that R(s)R(s) is an isomorphism in AA, then R:BAR \colon B \to A realizes BB as the (nonstrict) colocalization of BB with respect to the class SS.

  5. The right adjoint RR is codense.

For proofs, see the corresponding characterisations for reflective subcategories.

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 October 12, 2021 at 12:22:32. See the history of this page for a list of all contributions to it.