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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.
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.[ citation needed]
Dual notions to the above-mentioned notions are coreflection, coreflection arrow, (mono)coreflective subcategory, coreflective hull, anti-coreflective subcategory.
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![]() | This article includes a list of general
references, but it lacks sufficient corresponding
inline citations. (May 2015) |
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.
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.[ citation needed]
Dual notions to the above-mentioned notions are coreflection, coreflection arrow, (mono)coreflective subcategory, coreflective hull, anti-coreflective subcategory.
![]() | This section needs expansion. You can help by
adding to it. (April 2019) |
{{
cite book}}
: CS1 maint: multiple names: authors list (
link) CS1 maint: numeric names: authors list (
link)
{{
cite book}}
: CS1 maint: location missing publisher (
link)