# 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.^{[1]}^{: 91 } This adjoint is sometimes called a *reflector*, or *localization*.^{[2]} 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.

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.

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**.^{[citation needed]}

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