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{{Short description|Concept in mathematical theory of categories}}
{{More footnotes|date=May 2015}}
In [[mathematics]], a [[full subcategory]] ''A'' of a [[Category (mathematics)|category]] ''B'' is said to be '''reflective''' in ''B'' when the [[inclusion functor]] from ''A'' to ''B'' has a [[left adjoint]].{{r|Mac Lane|p=91}} This adjoint is sometimes called a ''reflector'', or ''localization''.<ref name="Rhiel">{{Cite book|title=Category theory in context|last=Riehl|first=Emily|author-link=Emily Riehl|isbn=9780486820804|location=Mineola, New York|oclc=976394474|page=140|date = 2017-03-09}}</ref> 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 (category theory)|object]] ''B'' there exists an '''A'''-object <math>A_B</math> and a '''B'''-[[morphism]] <math>r_B \colon B \to A_B</math> such that for each '''B'''-morphism <math>f\colon B\to A</math> to an '''A'''-object <math>A</math> there exists a unique '''A'''-morphism <math>\overline f \colon A_B \to A</math> with <math>\overline f\circ r_B=f</math>.

:[[File:Refl1.png]]

The pair <math>(A_B,r_B)</math> is called the '''A-reflection''' of ''B''. The morphism <math>r_B</math> is called the '''A-reflection arrow.''' (Although often, for the sake of brevity, we speak about <math>A_B</math> only as being the '''A'''-reflection of ''B'').

This is equivalent to saying that the embedding functor <math>E\colon \mathbf{A} \hookrightarrow \mathbf{B}</math> is a right adjoint. The left adjoint [[functor]] <math>R \colon \mathbf B \to \mathbf A</math> is called the '''reflector'''. The map <math>r_B</math> is the [[Adjoint functors#Unit and co-unit|unit]]{{Broken anchor|date=2024-11-07|bot=User:Cewbot/log/20201008/configuration|target_link=Adjoint functors#Unit and co-unit|reason= The anchor (Unit and co-unit) [[Special:Diff/180137064|has been deleted]].}} of this adjunction.

The reflector assigns to <math>B</math> the '''A'''-object <math>A_B</math> and <math>Rf</math> for a '''B'''-morphism <math>f</math> is determined by the [[Commutative diagram|commuting diagram]]

:[[File:Reflsq1.png]]

If all '''A'''-reflection arrows are (extremal) [[epimorphism]]s, then the subcategory '''A''' is said to be '''(extremal) epireflective'''. Similarly, it is '''bireflective''' if all reflection arrows are [[bimorphism]]s.

All these notions are special case of the common generalization&mdash;'''<math>E</math>-reflective subcategory,''' where <math>E</math> is a [[Class (set theory)|class]] of morphisms.

The '''<math>E</math>-reflective hull''' of a class '''A''' of objects is defined as the smallest <math>E</math>-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|date=May 2015}}

[[Dual (category theory)|Dual]] notions to the above-mentioned notions are coreflection, coreflection arrow, (mono)coreflective subcategory, coreflective hull, anti-coreflective subcategory.

==Examples==

===Algebra===
* The [[category of abelian groups]] '''Ab''' is a reflective subcategory of the [[category of groups]], '''Grp'''. The reflector is the functor that sends each group to its [[abelianization]]. In its turn, the category of groups is a reflective subcategory of the category of [[inverse semigroup]]s.<ref>Lawson (1998), [{{Google books|plainurl=y|id=2805q4tFiCkC|page=63|text=The category of groups is a reflective subcategory}} p. 63, Theorem 2.]</ref>
* Similarly, the category of [[Associative algebra|commutative associative algebras]] is a reflective subcategory of all associative algebras, where the reflector is [[Quotient_ring#For_algebras_over_a_ring|quotienting]] out by the commutator [[Ideal (ring theory)|ideal]]. This is used in the construction of the [[symmetric algebra]] from the [[tensor algebra]].
* Dually, the category of [[Anticommutativity|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 [[Field (mathematics)|fields]] is a reflective subcategory of the category of [[integral domain]]s (with [[injective]] [[ring homomorphism]]s as morphisms). The reflector is the functor that sends each integral domain to its [[field of fractions]].
* The category of abelian [[torsion group]]s is a coreflective subcategory of the category of abelian groups. The coreflector is the functor sending each group to its [[torsion subgroup]].
* The categories of [[elementary abelian group]]s, abelian [[p-group|''p''-groups]], and ''p''-groups are all reflective subcategories of the category of groups, and the [[Kernel (category theory)|kernels]] of the reflection maps are important objects of study; see [[focal subgroup theorem]].
* The [[category of groups]] is a coreflective subcategory of the category of [[monoid]]s: the right adjoint maps a monoid to its [[Unit (ring theory)|group of units]].<ref>{{Cite web|url=https://ncatlab.org/nlab/show/coreflective+subcategory|title=coreflective subcategory in nLab|website=ncatlab.org|access-date=2019-04-02}}</ref>

===Topology===
* The category of [[Kolmogorov space]]s (T<sub>0</sub> spaces) is a reflective subcategory of '''Top''', the [[category of topological spaces]], and the [[Kolmogorov quotient]] is the reflector.
*The category of [[completely regular space]]s '''CReg''' is a reflective subcategory of '''Top'''. By taking Kolmogorov quotients, one sees that the subcategory of [[Tychonoff space]]s is also reflective.
*The category of all [[compact space|compact]] [[Hausdorff space]]s is a reflective subcategory of the category of all Tychonoff spaces (and of the category of all topological spaces{{r|Rhiel|p=140}}). The reflector is given by the [[Stone–Čech compactification]].
*The category of all [[complete metric space]]s with [[Metric space#Uniformly continuous maps|uniformly continuous mappings]] is a reflective subcategory of the [[category of metric spaces]]. The reflector is the [[Completion (metric space)|completion]] of a metric space on objects, and the extension by density on arrows.{{r|Mac Lane|p=90}}
*The category of [[Sheaf (mathematics)|sheaves]] is a reflective subcategory of [[Presheaf (category theory)|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 space]]s is a coflective subcategory of '''Top'''.  The sequential coreflection of a topological space <math>(X,\tau)</math> is the space <math>(X,\tau_{\mathrm{seq}})</math>, where the topology <math>\tau_{\text{seq}}</math> is a finer topology than <math>\tau</math> consisting of all [[sequentially open]] sets in <math>X</math> (that is, complements of [[sequentially closed]] sets).{{sfn|Adámek|Herrlich|Strecker|2004|loc=Example 4.26 A(2)}}

===Functional analysis===
*The category of [[Banach spaces]] is a reflective subcategory of the category of [[normed space]]s and [[Bounded operator|bounded linear operators]]. The reflector is the norm completion functor.

===Category theory===
*For any [[Grothendieck site]] (''C'', ''J''), the [[topos]] of [[sheaf (mathematics)|sheaves]] on (''C'', ''J'') is a reflective subcategory of the topos of [[Presheaf (category theory)|presheaves]] on ''C'', with the special further property that the reflector functor is [[Exact functor|left exact]].  The reflector is the [[sheaf (mathematics)#Turning a presheaf into a sheaf|sheafification functor]] ''a'' : Presh(''C'') → Sh(''C'', ''J''), and the adjoint pair (''a'', ''i'') is an important example of a [[topos#Geometric morphisms|geometric morphism]] in topos theory.

==Properties==

{{Expand section|date=April 2019}}

* The components of the [[Counit (category theory)|counit]] are [[isomorphism]]s.{{r|Rhiel|p=140}}<ref name="Mac Lane">{{Cite book|title=Categories for the working mathematician|authorlink = Saunders Mac Lane|last=Mac Lane, Saunders, 1909-2005.|date=1998|publisher=Springer|isbn=0387984038|edition= 2nd|location=New York|oclc=37928530|page=89}}</ref>
* If ''D'' is a reflective subcategory of ''C'', then the inclusion functor ''D'' → ''C'' creates all [[Limit (category theory)|limits]] that are present in ''C''.{{r|Rhiel|p=141}}
* A reflective subcategory has all [[colimits]] that are present in the ambient category.{{r|Rhiel|p=141}}
* The [[Monad (category theory)|monad]] induced by the reflector/localization adjunction is idempotent.{{r|Rhiel|p=158}}

==Notes==
{{reflist}}

==References==
*{{cite book |last1=Adámek |first1=Jiří |last2=Herrlich |first2=Horst |last3=Strecker |first3=George E. | title=''Abstract and Concrete Categories''  | url=http://katmat.math.uni-bremen.de/acc/acc.pdf | publisher=[[John Wiley & Sons]]| location=New York | year=2004}}
* {{cite book
| author = [[Peter J. Freyd|Peter Freyd]], Andre Scedrov
| title = Categories, Allegories
| publisher = [[Elsevier|North-Holland]]
| series = Mathematical Library Vol 39
| year = 1990
| isbn = 978-0-444-70368-2 }}
*{{cite book | last=Herrlich | first=Horst | authorlink = Horst Herrlich | title=Topologische Reflexionen und Coreflexionen | publisher = [[Springer Science+Business Media|Springer]]| location=Berlin | year=1968  | others=Lecture Notes in Math. 78}}
* {{cite book|author=Mark V. Lawson|title=Inverse semigroups: the theory of partial symmetries|year=1998|publisher=World Scientific|isbn=978-981-02-3316-7}}

[[Category:Adjoint functors]]