In
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
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 ...
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.
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 (T
0 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