Reflective Subcategory
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In mathematics, a full subcategory ''A'' of a
category Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) ...
''B'' is said to be reflective in ''B'' when the inclusion functor from ''A'' to ''B'' has a
left adjoint In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are kno ...
. 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 In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are kno ...
. 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 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 In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and m ...
R \colon \mathbf B \to \mathbf A is called the reflector. The map r_B is the
unit Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or beat) in a theatrical presentation Music * ''Unit'' (a ...
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)
epimorphism In category theory, an epimorphism (also called an epic morphism or, colloquially, an epi) is a morphism ''f'' : ''X'' → ''Y'' that is right-cancellative in the sense that, for all objects ''Z'' and all morphisms , : g_1 \circ f = g_2 \circ f ...
s, then the subcategory A is said to be (extremal) epireflective. Similarly, it is bireflective if all reflection arrows are
bimorphism 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 ...
s. 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 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 In mathematics, the category Grp (or Gp) has the class of all groups for objects and group homomorphisms for morphisms. As such, it is a concrete category. The study of this category is known as group theory. Relation to other categories There a ...
, Grp. The reflector is the functor that sends each group to its
abelianization In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group. The commutator subgroup is important because it is the smallest normal ...
. 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 In mathematics, the symmetric algebra (also denoted on a vector space over a field is a commutative algebra over that contains , and is, in some sense, minimal for this property. Here, "minimal" means that satisfies the following universal ...
from the
tensor algebra In mathematics, the tensor algebra of a vector space ''V'', denoted ''T''(''V'') or ''T''(''V''), is the algebra of tensors on ''V'' (of any rank) with multiplication being the tensor product. It is the free algebra on ''V'', in the sense of being ...
. * 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 In mathematics, the exterior algebra, or Grassmann algebra, named after Hermann Grassmann, is an algebra that uses the exterior product or wedge product as its multiplication. In mathematics, the exterior product or wedge product of vectors is a ...
from the tensor algebra. * The category of
fields Fields may refer to: Music * Fields (band), an indie rock band formed in 2006 * Fields (progressive rock band), a progressive rock band formed in 1971 * ''Fields'' (album), an LP by Swedish-based indie rock band Junip (2010) * "Fields", a song b ...
is a reflective subcategory of the category of
integral domain In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural s ...
s (with injective
ring homomorphism In ring theory, a branch of abstract algebra, a ring homomorphism is a structure-preserving function between two rings. More explicitly, if ''R'' and ''S'' are rings, then a ring homomorphism is a function such that ''f'' is: :addition preser ...
s as morphisms). The reflector is the functor that sends each integral domain to its
field of fractions In abstract algebra, the field of fractions of an integral domain is the smallest field in which it can be embedded. The construction of the field of fractions is modeled on the relationship between the integral domain of integers and the field ...
. * The category of abelian
torsion group In group theory, a branch of mathematics, a torsion group or a periodic group is a group in which every element has finite order. The exponent of such a group, if it exists, is the least common multiple of the orders of the elements. For examp ...
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 groups, abelian ''p''-groups, and ''p''-groups are all reflective subcategories of the category of groups, and the
kernels Kernel may refer to: Computing * Kernel (operating system), the central component of most operating systems * Kernel (image processing), a matrix used for image convolution * Compute kernel, in GPGPU programming * Kernel method, in machine learnin ...
of the reflection maps are important objects of study; see
focal subgroup theorem In abstract algebra, the focal subgroup theorem describes the fusion of elements in a Sylow subgroup of a finite group. The focal subgroup theorem was introduced in and is the "first major application of the transfer" according to . The focal sub ...
. * The category of groups is a ''co''reflective subcategory of the category of
monoid In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0. Monoid ...
s: the right adjoint maps a monoid to its group of units.


Topology

* The category of
Kolmogorov space In topology and related branches of mathematics, a topological space ''X'' is a T0 space or Kolmogorov space (named after Andrey Kolmogorov) if for every pair of distinct points of ''X'', at least one of them has a neighborhood not containing the ...
s (T0 spaces) is a reflective subcategory of Top, the category of topological spaces, and the
Kolmogorov quotient In topology and related branches of mathematics, a topological space ''X'' is a T0 space or Kolmogorov space (named after Andrey Kolmogorov) if for every pair of distinct points of ''X'', at least one of them has a neighborhood not containing the ...
is the reflector. *The category of
completely regular space In topology and related branches of mathematics, Tychonoff spaces and completely regular spaces are kinds of topological spaces. These conditions are examples of separation axioms. A Tychonoff space refers to any completely regular space that is ...
s CReg is a reflective subcategory of Top. By taking Kolmogorov quotients, one sees that the subcategory of
Tychonoff space In topology and related branches of mathematics, Tychonoff spaces and completely regular spaces are kinds of topological spaces. These conditions are examples of separation axioms. A Tychonoff space refers to any completely regular space that is ...
s is also reflective. *The category of all
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...
Hausdorff space In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the m ...
s 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 In the mathematical discipline of general topology, Stone–Čech compactification (or Čech–Stone compactification) is a technique for constructing a universal map from a topological space ''X'' to a compact Hausdorff space ''βX''. The Ston ...
. *The category of all
complete metric space In mathematical analysis, a metric space is called complete (or a Cauchy space) if every Cauchy sequence of points in has a limit that is also in . Intuitively, a space is complete if there are no "points missing" from it (inside or at the bou ...
s with uniformly continuous mappings is a reflective subcategory of the
category of metric spaces In category theory, Met is a category that has metric spaces as its objects and metric maps (continuous functions between metric spaces that do not increase any pairwise distance) as its morphisms. This is a category because the composition of two ...
. 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 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 In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
is a reflective subcategory of the category of normed spaces and bounded linear operators. The reflector is the norm completion functor.


Category theory

*For any Grothendieck site (''C'', ''J''), the
topos 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 notio ...
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 functor ''a'' : Presh(''C'') → Sh(''C'', ''J''), and the adjoint pair (''a'', ''i'') is an important example of a geometric morphism in topos theory.


Properties

* The components of the counit are
isomorphism In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
s. * If ''D'' is a reflective subcategory of ''C'', then the inclusion functor ''D'' → ''C'' creates all
limits Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...
that are present in ''C''. * A reflective subcategory has all colimits that are present in the ambient category. * The
monad Monad may refer to: Philosophy * Monad (philosophy), a term meaning "unit" **Monism, the concept of "one essence" in the metaphysical and theological theory ** Monad (Gnosticism), the most primal aspect of God in Gnosticism * ''Great Monad'', a ...
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