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 $\overline f\circ r_B=f$.

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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 $E\colon \mathbf \hookrightarrow \mathbf$ 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

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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.

Examples

Algebra

* The category of abelian groups Ab is a reflective subcategory of the category of groups, Grp. The reflector is the functor that sends each group to its abelianization. In its turn, the category of groups is a reflective subcategory of the category of inverse semigroups.
* Similarly, the category of commutative associative algebras is a reflective subcategory of all associative algebras, where the reflector is quotienting out by the commutator ideal. This is used in the construction of the symmetric algebra from the tensor algebra.
* Dually, the category of anti-commutative associative algebras is a reflective subcategory of all associative algebras, where the reflector is quotienting out by the anti-commutator ideal. This is used in the construction of the exterior algebra from the tensor algebra.
* The category of fields is a reflective subcategory of the category of integral domains (with injective ring homomorphisms as morphisms). The reflector is the functor that sends each integral domain to its field of fractions.
* The category of abelian torsion groups is a coreflective subcategory of the category of abelian groups. The coreflector is the functor sending each group to its torsion subgroup.
* The categories of elementary abelian groups, abelian ''p''-groups, and ''p''-groups are all reflective subcategories of the category of groups, and the kernels of the reflection maps are important objects of study; see focal subgroup theorem.
* The category of groups is a coreflective subcategory of the category of monoids: the right adjoint maps a monoid to its group of units.

Topology

* The category of Kolmogorov spaces (T₀ spaces) is a reflective subcategory of Top, the category of topological spaces, and the Kolmogorov quotient is the reflector.
*The category of completely regular spaces CReg is a reflective subcategory of Top. By taking Kolmogorov quotients, one sees that the subcategory of Tychonoff spaces is also reflective.
*The category of all compact Hausdorff spaces is a reflective subcategory of the category of all Tychonoff spaces (and of the category of all topological spaces). The reflector is given by the Stone–Čech compactification.
*The category of all complete metric spaces with uniformly continuous mappings is a reflective subcategory of the category of metric spaces. The reflector is the completion of a metric space on objects, and the extension by density on arrows.
*The category of sheaves is a reflective subcategory of presheaves on a topological space. The reflector is sheafification, which assigns to a presheaf the sheaf of sections of the bundle of its germs.
*The category Seq of sequential spaces is a coreflective subcategory of Top. The sequential coreflection of a topological space $(X,\tau)$ is the space $(X,\tau_)$, where the topology $\tau_$ is a finer topology than $\tau$ consisting of all sequentially open sets in $X$ (that is, complements of sequentially closed sets).

Functional analysis

*The category of Banach spaces is a reflective subcategory of the category of normed spaces and bounded linear operators. The reflector is the norm completion functor.

Category theory

*For any Grothendieck site (''C'', ''J''), the topos of sheaves on (''C'', ''J'') is a reflective subcategory of the topos of presheaves on ''C'', with the special further property that the reflector functor is left exact. The reflector is the sheafification functor ''a'' : Presh(''C'') → Sh(''C'', ''J''), and the adjoint pair (''a'', ''i'') is an important example of a geometric morphism in topos theory.

Properties

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

Notes

References

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* {{cite book, author=Mark V. Lawson, title=Inverse semigroups: the theory of partial symmetries, year=1998, publisher=World Scientific, isbn=978-981-02-3316-7

Adjoint functors