# Reflective subcategory

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.

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

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

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

The reflector assigns to ${\displaystyle B}$ the A-object ${\displaystyle A_{B}}$ and ${\displaystyle Rf}$ for a B-morphism ${\displaystyle f}$ 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 case of the common generalization — ${\displaystyle E}$-reflective subcategory, where ${\displaystyle E}$ is a class of morphisms.

The ${\displaystyle E}$-reflective hull of a class A of objects is defined as the smallest ${\displaystyle E}$-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.

## Notes

1. Lawson (1998), [[[:Template:Google books]] p. 63, Theorem 2.]

## References

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