nLab coreflective subcategory

Definition

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

$$C \stackrel{\overset{i}{\hookrightarrow}}{\underset{R}{\leftarrow}} D \,.$$

The dual concept is that of a reflective subcategory. See there for more details.

Characterizations

Proposition

(equivalent characterizations)

Given any pair of adjoint functors

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

the following are equivalent:

1. The left adjoint $L$ is fully faithful. (In this case $A$ is equivalent to its essential image in $B$ under $L$, a full coreflective subcategory of $B$.)

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

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

4. If $S$ is the set of morphisms $s$ in $B$ such that $R(s)$ is an isomorphism in $A$, then $R \colon B \to A$ realizes $B$ as the (nonstrict) colocalization of $B$ with respect to the class $S$.

5. The right adjoint $R$ is codense.

For proofs, see the corresponding characterisations for reflective subcategories.

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 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 $G$ from the universal covering space of the connected component at the identity of $G$.

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

Related concepts

• 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]