Reflective subcategory

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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.

Contents 1 Definition 2 Examples 2.1 Algebra 2.2 Topology 2.3 Category theory 3 References

[edit] 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 A_B and a morphism such that for each B-morphism there exists a unique A-morphism with .

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

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.

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

[edit] Examples

[edit] 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.

• 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 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.

[edit] Topology

• Kolmogorov spaces (T₀ spaces) are a reflective subcategory of all topological spaces, and the Kolmogorov quotient is the reflector.

• The category of all compact Hausdorff spaces is a reflective subcategory of the category of all topological spaces. The reflector is given by the Stone-Čech compactification.

[edit] 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. 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.

[edit] References

• Adámek, Jiří; Horst Herrlich, George E. Strecker (1990). Abstract and Concrete Categories. New York: John Wiley & Sons. 

• Herrlich, Horst (1968). Topologische Reflexionen und Coreflexionen, Lecture Notes in Math. 78, Berlin: Springer.