coreflective subcategory in nLab

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Contents

Definition

A coreflective subcategory is a full subcategory whose inclusion functor has a right adjoint RR (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.

Characterizations

For proofs, see the corresponding characterisations for reflective subcategories.

Properties

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 *:A′↪Ai_*: A' \hookrightarrow A with reflector i *i^* and coreflector i !i^!. The functor j *j^* is both a reflector for the reflective subcategory j *:A″↪Aj_*: A'' \hookrightarrow A, and a coreflector for the coreflective subcategory j !:A″↪Aj_!: A'' \hookrightarrow A.

• Quillen coreflection

• reflective subcategory

• bireflective subcategory

References

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

• Robert El Bashir, Jiri Velebil, Simultaneously Reflective And Coreflective Subcategories of Presheaves (TAC)

• Michael Barr, John Kennison, Robert Raphael, On reflective and coreflective hulls, Cahiers Topologie Géométrie Différentielle Catégorique 56 (2015) 162–208 [pdf, pdf]