Reflective subcategory

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In mathematics, a 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. Dually, A is said to be coreflective in B when the inclusion functor has a right adjoint.

Definition[edit]

A 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 and a B-morphism r_B \colon B \to A_B such that for each B-morphism f\colon B\to A there exists a unique A-morphism \overline f \colon A_B \to A with \overline f\circ r_B=f.

Refl1.png

The pair (A_B,r_B) is called the A-reflection of B. The morphism r_B is called A-reflection arrow. (Although often, for the sake of brevity, we speak about A_B only as about the A-reflection of B).

This is equivalent to saying that the embedding functor E\colon \mathbf{A} \hookrightarrow \mathbf{B} is adjoint. The coadjoint functor R \colon \mathbf B \to \mathbf A is called the reflector. The map r_B is the unit of this adjunction.

The reflector assigns to B the A-object A_B and Rf for a B-morphism f is determined by the commuting diagram

Reflsq1.png

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 case of the common generalization — E-reflective subcategory, where E is a class of morphisms.

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

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

Examples[edit]

Algebra[edit]

Topology[edit]

Functional analysis[edit]

Category theory[edit]

Notes[edit]

  1. ^ Lawson (1998), p. 63, Theorem 2.

References[edit]