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.[1]: 91  This adjoint is sometimes called a reflector, or localization.[2] 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 ${\displaystyle A_{B}}$ and a B-morphism ${\displaystyle r_{B}\colon B\to A_{B}}$ such that for each B-morphism ${\displaystyle f\colon B\to A}$ to an A-object ${\displaystyle A}$ 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 the A-reflection arrow. (Although often, for the sake of brevity, we speak about ${\displaystyle A_{B}}$ only as being the A-reflection of B).

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

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.

Properties

• The components of the counit are isomorphisms.[2]: 140 [1]
• If D is a reflective subcategory of C, then the inclusion functor DC creates all limits that are present in C.[2]: 141
• A reflective subcategory has all colimits that are present in the ambient category.[2]: 141
1. ^ a b c Mac Lane, Saunders, 1909-2005. (1998). Categories for the working mathematician (2nd ed.). New York: Springer. p. 89. ISBN 0387984038. OCLC 37928530.{{cite book}}: CS1 maint: multiple names: authors list (link) CS1 maint: numeric names: authors list (link)
2. Riehl, Emily (2017-03-09). Category theory in context. Mineola, New York. p. 140. ISBN 9780486820804. OCLC 976394474.{{cite book}}: CS1 maint: location missing publisher (link)