A subcategory which contains a
"largest"
model of any
object of a given category. More precisely, a full subcategory
of a category
is called
reflective
if it contains a reflection (cf.
Reflection of an object of a category)
for every object of
.
Equivalently,
is reflective in
if and only if the inclusion functor
has a left adjoint
.
The functor
sends each object
of
to its
-reflection
;
the morphisms
appearing in the definition of a reflection constitute
a natural transformation from the identity functor on
to the composite of
with the inclusion functor, which is the unit of the adjunction (see
Adjoint functor).
The concept dual to that of a reflective subcategory is called a
coreflective subcategory.
A reflective subcategory
inherits many properties from the ambient category
.
For example, a morphism
of
is a
monomorphism
in
if and only if it is a monomorphism in
.
Therefore, every reflective subcategory of a well-powered
category is well-powered. A reflective subcategory is closed
under products, to the extent that they exist in
the ambient category. The same holds for more general
limits. A reflective subcategory need not be closed under colimits, but the functor
transforms colimits in
into colimits in
.
Thus, a reflective subcategory of a
complete (cocomplete) category is complete (cocomplete).
Suppose
is complete and has a
bicategory
(factorization) structure in which every object has only a
set of admissible quotients. Then every full subcategory
of
which is closed under products and admissible subobjects
is reflective. In this context, one may construct the
-reflection
of an object
of
as follows: Choose a set of representatives
,
,
of those quotient objects of
which lie in
.
The product
belongs to
,
and the
-reflection
is the image of the unique morphism
such that
,
.
Examples.
1)
Let
be an integral domain. The full subcategory
of torsion-free injective modules is reflective in the category of all torsion-free
-modules;
the reflections are the injective hulls of torsion-free modules. In
particular, the full subcategory of divisible torsion-free Abelian groups
is reflective in the category of torsion-free Abelian groups.
2)
The full subcategory of compact Hausdorff topological spaces
is reflective in the category of completely
regular topological spaces. The Stone–Čech compactification provides the reflector.
3)
The category of sheaves on a topological space is
reflective in the category of pre-sheaves. The reflector is
defined by the associated sheaf functor
(sheafication).