# Reflective subcategory

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 $A_{B}$ and a B-morphism $r_{B}\colon B\to A_{B}$ such that for each B-morphism $f\colon B\to A$ to an A-object $A$ there exists a unique A-morphism ${\overline {f}}\colon A_{B}\to A$ with ${\overline {f}}\circ r_{B}=f$ .

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

This is equivalent to saying that the embedding functor $E\colon \mathbf {A} \hookrightarrow \mathbf {B}$ is a right adjoint. The left adjoint 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

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.

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.

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

## Properties

• The components of the counit are isomorphisms.:140
• If D is a reflective subcategory of C, then the inclusion functor DC creates all limits that are present in C.:141
• A reflective subcategory has all colimits that are present in the ambient category.:141
• The monad induced by the reflector/localization adjunction is idempotent.:158