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

Topology

Functional analysis

Category theory

Properties

  • The components of the counit are isomorphisms.[1]:140[4]
  • If D is a reflective subcategory of C, then the inclusion functor DC 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

  1. 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)
  2. Lawson (1998), p. 63, Theorem 2.
  3. "coreflective subcategory in nLab". ncatlab.org. Retrieved 2019-04-02.
  4. 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

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