Reflective subcategory

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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. 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[edit]

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 and a B-morphism such that for each B-morphism to an A-object there exists a unique A-morphism with .

Refl1.png

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

This is equivalent to saying that the embedding functor is adjoint. The coadjoint functor is called the reflector. The map is the unit of this adjunction.

The reflector assigns to the A-object and for a B-morphism 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 — -reflective subcategory, where is a class of morphisms.

The -reflective hull of a class A of objects is defined as the smallest -reflective subcategory containing A. Thus we can speak about 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[edit]

Algebra[edit]

Topology[edit]

Functional analysis[edit]

Category theory[edit]

Notes[edit]

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

References[edit]