# 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*.[1] 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 and a **B**-morphism such that for each **B**-morphism to an **A**-object there exists a unique **A**-morphism with .

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

This is equivalent to saying that the embedding functor is a right adjoint. The left adjoint 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

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**.

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 which sends each group to its abelianization. In its turn, the category of groups is a reflective subcategory of the category of inverse semigroups.[2] - 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 which 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
*co*reflective subcategory of the category of monoids: the right adjoint maps a monoid to its group of units.[3]

### Topology

- Kolmogorov spaces (T
_{0}spaces) are 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[1]
^{:140}). 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.[4]
^{:90}

### 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.[1]
^{:140}[4] - If
*D*is a reflective subcategory of*C*, then the inclusion functor*D*→*C*creates all limits that are present in*C*.[1]^{:141} - A reflective subcategory has all colimits that are present in the ambient category.[1]
^{:141} - The monad induced by the reflector/localization adjunction is idempotent.[1]
^{:158}

## Notes

- Riehl, Emily,.
*Category theory in context*. Mineola, New York. p. 140. ISBN 9780486820804. OCLC 976394474.CS1 maint: extra punctuation (link) CS1 maint: multiple names: authors list (link) - Lawson (1998), p. 63, Theorem 2.
- "coreflective subcategory in nLab".
*ncatlab.org*. Retrieved 2019-04-02. - Mac Lane, Saunders, 1909-2005. (1998).
*Categories for the working mathematician*(2nd ed.). New York: Springer. p. 89. ISBN 0387984038. OCLC 37928530.CS1 maint: multiple names: authors list (link)

## References

- Adámek, Jiří; Horst Herrlich; George E. Strecker (1990). Abstract and Concrete Categories (PDF). New York: John Wiley & Sons.
- Peter Freyd, Andre Scedrov (1990).
*Categories, Allegories*. Mathematical Library Vol 39. North-Holland. ISBN 978-0-444-70368-2. - Herrlich, Horst (1968).
*Topologische Reflexionen und Coreflexionen*. Lecture Notes in Math. 78. Berlin: Springer. - Mark V. Lawson (1998).
*Inverse semigroups: the theory of partial symmetries*. World Scientific. ISBN 978-981-02-3316-7.