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-
objectB 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.[citation needed]
Dual notions to the above-mentioned notions are coreflection, coreflection arrow, (mono)coreflective subcategory, coreflective hull, anti-coreflective subcategory.
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.
The category of
completely regular spacesCReg 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
compactHausdorff spaces is a reflective subcategory of the category of all Tychonoff spaces (and of the category of all topological spaces[2]: 140 ). The reflector is given by the
Stone–Čech compactification.
The category of
sheaves is a reflective subcategory of
presheaves on a topological space. The reflector is sheafification, which assigns to a presheaf the sheaf of sections of the bundle of its germs.
The category Seq of
sequential spaces is a coflective subcategory of Top. The sequential coreflection of a topological space is the space , where the topology is a finer topology than consisting of all
sequentially open sets in (that is, complements of
sequentially closed sets).[5]
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 functora : Presh(C) → Sh(C, J), and the adjoint pair (a, i) is an important example of a
geometric morphism in topos theory.
Properties
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adding to it. (April 2019)