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| In [[mathematics]], a [[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]]. This adjoint is sometimes called a ''reflector''. Dually, ''A'' is said to be '''coreflective''' in ''B'' when the inclusion functor has a [[right adjoint]].
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| ==Definition==
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| A subcategory '''A''' of a category '''B''' is said to be '''reflective in B''' if for each '''B'''-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> there exists a unique '''A'''-morphism <math>\overline f \colon A_B \to A</math> with <math>\overline f\circ r_B=f</math>.
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| :[[File:Refl1.png]]
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| The pair <math>(A_B,r_B)</math> is called the '''A-reflection''' of ''B''. The morphism <math>r_B</math> is called '''A-reflection arrow.''' (Although often, for the sake of brevity, we speak about <math>A_B</math> only as about the '''A'''-reflection of ''B''. | |
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| This is equivalent to saying that the embedding functor <math>E\colon \mathbf{A} \hookrightarrow \mathbf{B}</math> is adjoint. The coadjoint 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]] of this adjunction.
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| 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
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| the commuting diagram
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| :[[File:Reflsq1.png]]
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| 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.
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| All these notions are special case of the common generalization — '''<math>E</math>-reflective subcategory,''' where <math>E</math> is a class of morphisms.
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| 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.
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| [[Dual (category theory)|Dual]] notions to the above mentioned notions are coreflection, coreflection arrow, (mono)coreflective subcategory, coreflective hull.
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| ==Examples==
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| ===Algebra===
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| * The [[category of abelian groups]] '''Ab''' is a reflective subcategory of the [[category of groups]], '''Grp'''. The reflector is the functor which 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 quote|id=2805q4tFiCkC|page=63|text=The category of groups is a reflective subcategory|p. 63, Theorem 2.}}</ref>
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| * Similarly, the category of commutative [[associative algebra]]s is a reflective subcategory of all associative algebras, where the reflector is [[Quotient algebra|quotienting]] out by the commutator [[Ideal (ring theory)|ideal]]. This is used in the construction of the [[symmetric algebra]] from the [[tensor algebra]].
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| * 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]].
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| * The category of [[Field (mathematics)|fields]] is a reflective subcategory of the category of [[integral domain]]s (with [[injective]] ring homomorphisms as morphisms). The reflector is the functor which sends each integral domain to its [[field of fractions]].
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| * 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]].
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| * The categories of [[elementary abelian group]]s, abelian ''p''-groups, and ''p''-groups are all reflective subcategories of the category of groups, and the kernels of the reflection maps are important objects of study; see [[focal subgroup theorem]].
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| *The [[category of vector spaces]] over the field ''k'' is a (non full) reflective subcategory of the category of sets. The reflector is the functor which sends each set B in the [[Free module|free vector space]] generated by B over ''k'', that can be identified with the vector space of all ''k'' valued functions on B vanishing outside a finite set. In similar way, several free construction functors are reflectors of the category of sets onto the corresponding reflective subcategory.
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| ===Topology===
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| * [[Kolmogorov space]]s (T<sub>0</sub> spaces) are a reflective subcategory of '''Top''', the [[category of topological spaces]], and the [[Kolmogorov quotient]] is the reflector.
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| *The category of [[completely regular space]]s '''CReg''' is a reflective subcategory of '''Top'''. By taking [[Kolmogorov quotient]]s, one sees that the subcategory of [[Tychonoff space]]s is also reflective.
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| *The category of all [[compact space|compact]] [[Hausdorff space]]s is a reflective subcategory of the category of all Tychonoff spaces. The reflector is given by the [[Stone–Čech compactification]].
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| *The category of all [[complete metric space]]s with [[metric space#Uniformly continuous maps|uniformly continuous mappings]] is a reflective and full 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.
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| ===Functional analysis===
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| *The category of [[Banach spaces]] is a reflective and full subcategory of the category of [[normed space]]s and [[Bounded operator|bounded linear operators]]. The reflector is the norm completion functor.
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| ===Category theory===
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| *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.
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| ==Notes==
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| {{reflist}}
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| ==References==
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| *{{cite book | last=Adámek | first=Jiří | coauthors=Horst Herrlich, George E. Strecker | title=''Abstract and Concrete Categories'' | url=http://katmat.math.uni-bremen.de/acc/acc.pdf | publisher=[[John Wiley & Sons]]| location=New York | year=1990}}
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| * {{cite book
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| | author = [[Peter J. Freyd|Peter Freyd]], Andre Scedrov
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| | title = Categories, Allegories
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| | publisher = [[Elsevier|North-Holland]]
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| | series = Mathematical Library Vol 39
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| | year = 1990
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| | isbn = 978-0-444-70368-2 }}
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| *{{cite book | last=Herrlich | first=Horst | title=Topologische Reflexionen und Coreflexionen | publisher = [[Springer Science+Business Media|Springer]]| location=Berlin | year=1968 | others=Lecture Notes in Math. 78}}
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| * {{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}}
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| [[Category:Adjoint functors]]
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The person who wrote the article is known as Jayson Hirano and he totally digs that name. I've always cherished living in Alaska. Office supervising is exactly where my primary income arrives from but I've always needed my own company. The preferred hobby for him and his kids is to play lacross and he would never give it up.
My site ... psychic phone