# 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 and a **B**-morphism such that for each **B**-morphism there exists a unique **A**-morphism with .

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

This is equivalent to saying that the embedding functor is adjoint. The coadjoint 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.

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

## 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.^{[1]} - 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 vector spaces over the field
*k*is a (non full) reflective subcategory of the category of sets. The reflector is the functor which sends each set B in the free vector space generated by B over*k*, that can be identified with the vector space of all*k*valued functions on B vanishing outside a finite set. In similar way, several free construction functors are reflectors of the category of sets onto the corresponding reflective subcategory.

### 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. The reflector is given by the Stone–Čech compactification.
- The category of all complete metric spaces with uniformly continuous mappings is a reflective and full 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.

### Functional analysis

- The category of Banach spaces is a reflective and full 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.

## Notes

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

## References

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