In mathematics, a full subcategory A of a category B is said to be reflective in B when the inclusion functor from A to B has a left adjoint. This adjoint is sometimes called a reflector, or localization. Dually, A is said to be coreflective in B when the inclusion functor has a right adjoint.

Informally, a reflector acts as a kind of completion operation. It adds in any "missing" pieces of the structure in such a way that reflecting it again has no further effect.

Definition

A full subcategory A of a category B is said to be reflective in B if for each B-object B there exists an A-object A B {\displaystyle A_{B}} and a B-morphism r B : B → A B {\displaystyle r_{B}\colon B\to A_{B}} such that for each B-morphism f : B → A {\displaystyle f\colon B\to A} to an A-object A {\displaystyle A} there exists a unique A-morphism f ¯ : A B → A {\displaystyle {\overline {f}}\colon A_{B}\to A} with f ¯ ∘ r B = f {\displaystyle {\overline {f}}\circ r_{B}=f}.

The pair ( A B , r B ) {\displaystyle (A_{B},r_{B})} is called the A-reflection of B. The morphism r B {\displaystyle r_{B}} is called the A-reflection arrow. (Although often, for the sake of brevity, we speak about A B {\displaystyle A_{B}} only as being the A-reflection of B).

This is equivalent to saying that the embedding functor E : A ↪ B {\displaystyle E\colon \mathbf {A} \hookrightarrow \mathbf {B} } is a right adjoint. The left adjoint functor R : B → A {\displaystyle R\colon \mathbf {B} \to \mathbf {A} } is called the reflector. The map r B {\displaystyle r_{B}} is the unit of this adjunction.

The reflector assigns to B {\displaystyle B} the A-object A B {\displaystyle A_{B}} and R f {\displaystyle Rf} for a B-morphism f {\displaystyle f} is determined by the commuting diagram

If all A-reflection arrows are (extremal) epimorphisms, then the subcategory A is said to be (extremal) epireflective. Similarly, it is bireflective if all reflection arrows are bimorphisms.

All these notions are special cases of the common generalization E {\displaystyle E}-reflective subcategory, where E {\displaystyle E} is a class of morphisms.

The E {\displaystyle E}-reflective hull of a class A of objects is defined as the smallest E {\displaystyle E}-reflective subcategory containing A. Thus we can speak about the reflective hull, epireflective hull, extremal epireflective hull, etc.

An anti-reflective subcategory is a full subcategory A such that the only objects of B that have an A-reflection arrow are those that are already in A.[citation needed]

Dual notions to the above-mentioned notions are coreflection, coreflection arrow, (mono)coreflective subcategory, coreflective hull, anti-coreflective subcategory.

Examples

Algebra

Topology

  • The category of Kolmogorov spaces (T0 spaces) is a reflective subcategory of Top, the category of topological spaces, and the Kolmogorov quotient is the reflector.
  • The category of completely regular spaces CReg is a reflective subcategory of Top. By taking Kolmogorov quotients, one sees that the subcategory of Tychonoff spaces is also reflective.
  • The category of all compact Hausdorff spaces is a reflective subcategory of the category of all Tychonoff spaces (and of the category of all topological spaces). The reflector is given by the Stone–Čech compactification.
  • The category of all complete metric spaces with uniformly continuous mappings is a reflective subcategory of the category of metric spaces. The reflector is the completion of a metric space on objects, and the extension by density on arrows.
  • The category of sheaves is a reflective subcategory of presheaves on a topological space. The reflector is sheafification, which assigns to a presheaf the sheaf of sections of the bundle of its germs.
  • The category Seq of sequential spaces is a coreflective subcategory of Top. The sequential coreflection of a topological space ( X , τ ) {\displaystyle (X,\tau )} is the space ( X , τ s e q ) {\displaystyle (X,\tau _{\mathrm {seq} })}, where the topology τ seq {\displaystyle \tau _{\text{seq}}} is a finer topology than τ {\displaystyle \tau } consisting of all sequentially open sets in X {\displaystyle X} (that is, complements of sequentially closed sets).

Functional analysis

Category theory

  • For any Grothendieck site (C, J), the topos of sheaves on (C, J) is a reflective subcategory of the topos of presheaves on C, with the special further property that the reflector functor is left exact. The reflector is the sheafification functor a: Presh(C) → Sh(C, J), and the adjoint pair (a, i) is an important example of a geometric morphism in topos theory.

Properties

  • The components of the counit are isomorphisms.
  • If D is a reflective subcategory of C, then the inclusion functor D → C creates all limits that are present in C.
  • A reflective subcategory has all colimits that are present in the ambient category.
  • The monad induced by the reflector/localization adjunction is idempotent.

Notes

  • Adámek, Jiří; Herrlich, Horst; Strecker, George E. (2004). (PDF). New York: John Wiley & Sons.
  • Peter Freyd, Andre Scedrov (1990). Categories, Allegories. Mathematical Library Vol 39. North-Holland. ISBN978-0-444-70368-2.
  • Herrlich, Horst (1968). Topologische Reflexionen und Coreflexionen. Lecture Notes in Math. 78. Berlin: Springer.
  • Mark V. Lawson (1998). Inverse semigroups: the theory of partial symmetries. World Scientific. ISBN978-981-02-3316-7.