Reflective subcategory

A subcategory which contains a "largest" model of any object of a given category. More precisely, a full subcategory $\mathfrak{C}$ of a category $\mathfrak{K}$ is called reflective if it contains a reflection (cf. Reflection of an object of a category) for every object of $\mathfrak{K}$. Equivalently, $\mathfrak{C}$ is reflective in $\mathfrak{K}$ if and only if the inclusion functor $\mathfrak{C}\rightarrow\mathfrak{K}$ has a left adjoint $S:\mathfrak{K}\rightarrow\mathfrak{C}$. The functor $S$ sends each object $A$ of $\mathfrak{K}$ to its $\mathfrak{C}$-reflection $S(A)$; the morphisms $\pi_A : A \rightarrow S(A)$ appearing in the definition of a reflection constitute a natural transformation from the identity functor on $\mathfrak{K}$ to the composite of $S$ 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 $\mathfrak{C}$ inherits many properties from the ambient category $\mathfrak{K}$. For example, a morphism $\mu$ of $\mathfrak{C}$ is a monomorphism in $\mathfrak{C}$ if and only if it is a monomorphism in $\mathfrak{K}$. 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 $S$ transforms colimits in $\mathfrak{K}$ into colimits in $\mathfrak{C}$. Thus, a reflective subcategory of a complete (cocomplete) category is complete (cocomplete).

Suppose $\mathfrak{K}$ is complete and has a bicategory (factorization) structure in which every object has only a set of admissible quotients. Then every full subcategory $\mathfrak{C}$ of $\mathfrak{K}$ which is closed under products and admissible subobjects is reflective. In this context, one may construct the $\mathfrak{C}$-reflection of an object $A$ of $\mathfrak{K}$ as follows: Choose a set of representatives $\{\gamma_i:A\rightarrow A_i\}$, $i \in I$, of those quotient objects of $A$ which lie in $\mathfrak{C}$. The product $P = \prod_{i\in I} A_i$ belongs to $\mathfrak{C}$, and the $\mathfrak{C}$-reflection $S(A)$ is the image of the unique morphism $\gamma : A \rightarrow P$ such that $\pi_i \gamma = \gamma_i$, $i\in I$.

Examples.

1) Let $R$ be an integral domain. The full subcategory of torsion-free injective modules is reflective in the category of all torsion-free $R$-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): cf. Sheaf theory.

Comments

Some authors (such as Mac Lane [a1]) extend the use of the term "reflective subcategory" to include non-full subcategories for which the inclusion functor has a left adjoint.

A reflective subcategory is called epireflective if the canonical morphism $\pi_A : A \rightarrow S(A)$ is an epimorphism for every $A$. If every morphism in $\mathfrak{K}$ factors as an epimorphism followed by a monomorphism, then a reflective subcategory of $\mathfrak{K}$ will be epireflective provided it is closed under arbitrary subobjects in $\mathfrak{K}$. The three examples listed in the main article are not epireflective, but (for example) the category of Abelian groups is epireflective in the category of all groups. The dual concept is that of a monocoreflective subcategory; for example, the category of torsion Abelian groups is monocoreflective in the category of all Abelian groups.

References

[a1]	Saunders Mac Lane, Categories for the Working Mathematician (2 ed) Graduate Texts in Mathematics 5, Springer (1998) ISBN 0-387-98403-8 Zbl 0906.18001