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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a full subcategory ''A'' of a category ''B'' is said to be reflective in ''B'' when the
inclusion functor In mathematics, specifically category theory, a subcategory of a category ''C'' is a category ''S'' whose objects are objects in ''C'' and whose morphisms are morphisms in ''C'' with the same identities and composition of morphisms. Intuitivel ...
from ''A'' to ''B'' has a left adjoint. This adjoint is sometimes called a ''reflector'', or ''localization''. 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 ''B'' there exists an A-object A_B and a B-
morphism In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms a ...
r_B \colon B \to A_B such that for each B-morphism f\colon B\to A to an A-object A there exists a unique A-morphism \overline f \colon A_B \to A with \overline f\circ r_B=f. : The pair (A_B,r_B) is called the A-reflection of ''B''. The morphism r_B is called the A-reflection arrow. (Although often, for the sake of brevity, we speak about A_B only as being the A-reflection of ''B''). This is equivalent to saying that the embedding functor E\colon \mathbf \hookrightarrow \mathbf is a right adjoint. The left adjoint functor R \colon \mathbf B \to \mathbf A is called the reflector. The map r_B is the unit of this adjunction. The reflector assigns to B the A-object A_B and Rf for a B-morphism f 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—E-reflective subcategory, where E is a class of morphisms. The E-reflective hull of a class A of objects is defined as the smallest E-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.
Dual Dual or Duals may refer to: Paired/two things * Dual (mathematics), a notion of paired concepts that mirror one another ** Dual (category theory), a formalization of mathematical duality *** see more cases in :Duality theories * Dual (grammatical ...
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 In mathematics, the category Ab has the abelian groups as objects and group homomorphisms as morphisms. This is the prototype of an abelian category: indeed, every small abelian category can be embedded in Ab. Properties The zero object of Ab is ...
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 semigroups. * Similarly, the category of commutative associative algebras is a reflective subcategory of all associative algebras, where the reflector is quotienting out by the commutator ideal. This is used in the construction of the symmetric algebra from the tensor algebra. * Dually, the category of
anti-commutative In mathematics, anticommutativity is a specific property of some non-commutative mathematical operations. Swapping the position of two arguments of an antisymmetric operation yields a result which is the ''inverse'' of the result with unswapped ...
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 fields is a reflective subcategory of the category of integral domains (with
injective In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositiv ...
ring homomorphisms as morphisms). The reflector is the functor that sends each integral domain to its field of fractions. * 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 categories of elementary abelian groups, 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. * The category of groups is a ''co''reflective subcategory of the category of monoids: the right adjoint maps a monoid to its
group of units In algebra, a unit of a ring is an invertible element for the multiplication of the ring. That is, an element of a ring is a unit if there exists in such that vu = uv = 1, where is the multiplicative identity; the element is unique for this ...
.


Topology

* The category of Kolmogorov spaces (T0 spaces) is a reflective subcategory of Top, the
category of topological spaces In mathematics, the category of topological spaces, often denoted Top, is the category whose objects are topological spaces and whose morphisms are continuous maps. This is a category because the composition of two continuous maps is again contin ...
, and the Kolmogorov quotient is the reflector. *The category of completely regular spaces CReg 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 compact Hausdorff spaces is a reflective subcategory of the category of all Tychonoff spaces (and of the category of all topological spaces). The reflector is given by the Stone–Čech compactification. *The category of all complete metric spaces with uniformly continuous mappings is a reflective subcategory of the category of metric spaces. The reflector is the completion of a metric space on objects, and the extension by density on arrows. *The category of sheaves is a reflective subcategory of
presheaves In mathematics, a sheaf is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open sets of a topological space and defined locally with regard to them. For example, for each open set, the data could ...
on a topological space. The reflector is sheafification, which assigns to a presheaf the sheaf of sections of the bundle of its germs.


Functional analysis

*The category of Banach spaces is a reflective subcategory of the category of
normed space In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length" i ...
s and bounded linear operators. The reflector is the norm completion functor.


Category theory

*For any Grothendieck site (''C'', ''J''), the topos of sheaves on (''C'', ''J'') is a reflective subcategory of the topos of
presheaves In mathematics, a sheaf is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open sets of a topological space and defined locally with regard to them. For example, for each open set, the data could ...
on ''C'', with the special further property that the reflector functor is left exact. The reflector is the sheafification functor ''a'' : Presh(''C'') → Sh(''C'', ''J''), and the adjoint pair (''a'', ''i'') is an important example of a
geometric morphism In mathematics, a topos (, ; plural topoi or , or toposes) is a category that behaves like the category of sheaves of sets on a topological space (or more generally: on a site). Topoi behave much like the category of sets and possess a notion ...
in topos theory.


Properties

* The components of the
counit In mathematics, coalgebras or cogebras are structures that are dual (in the category-theoretic sense of reversing arrows) to unital associative algebras. The axioms of unital associative algebras can be formulated in terms of commutative diagram ...
are isomorphisms. * If ''D'' is a reflective subcategory of ''C'', then the inclusion functor ''D'' → ''C'' creates all limits that are present in ''C''. * A reflective subcategory has all colimits that are present in the ambient category. * The monad induced by the reflector/localization adjunction is idempotent.


Notes


References

* * * * {{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 Adjoint functors