The left adjoint is sometimes called the reflector, and a functor which is a reflector (or has a fully faithful right adjoint, which is the same up to equivalence) is called a reflection. Of course, there are dual notions of coreflective subcategory, coreflector, and coreflection.
A few sources (such as Categories Work) do not require a reflective subcategory to be full. However, in light of the fact that non-full subcategories are not invariant under equivalence, consideration of non-full reflective subcategories seems of limited usefulness. The general consensus among category theorists nowadays seems to be that “reflective subcategory” implies fullness.
The components of the unit
of this adjunction “reflect” each object into its image in the reflective subcategory
Given any pair of adjoint functors
the following are equivalent:
If is the set of morphisms in such that is invertible in , then realizes as the (nonstrict) localization of with respect to the class .
This is due to Gabriel-Zisman, (proposition 1.3, page 7).
This is a well-known set of equivalences concerning idempotent monads. The essential point is that a reflective subcategory is monadic, i.e., realizes as the category of algebras for the monad on , where is the reflector.
See also the related discussion at reflective sub-(infinity,1)-category.
If the reflector (which as a left adjoint always preserves all colimits) in addition preserves finite limits, then the embedding is called exact . If the categories are toposes then such embeddings are called geometric embeddings.
The reflector in that case is the sheafification functor.
In particular, is then also cartesian closed.
So in particular if is an exact reflective subcategory of a cartesian closed category , then is an exponential ideal of .
See Day's reflection theorem for a more general statement and proof.
When the unit of the reflector is a monomorphism, a reflective category is often thought of as a full subcategory of complete objects in some sense; the reflector takes each object in the ambient category to its completion. Such reflective subcategories are sometimes called mono-reflective. One similarly has epi-reflective (when the unit is an epimorphism) and bi-reflective (when the unit is a bimorphism).
In the last case, note that if the unit is an isomorphism, then the inclusion functor is an equivalence of categories, so nontrivial bireflective subcategories can occur only in non-balanced categories. Also note that ‘bireflective’ here does not mean reflective and coreflective. One sees this term often in discussions of concrete categories (such as topological categories) where really something stronger holds: that the reflector lies over the identity functor on Set. In this case, one can say that we have a subcategory that is reflective over .
In this explicit form this appears as (Lurie, prop. 22.214.171.124). From (Adamek-Rosický) the “only if”-direction follows immediately from 2.53 there (saying that an accessibly embedded subcategory of an accessible category is accessible iff it is cone-reflective), while “if”-direction follows immediately from 2.23 (saying any left or right adjoint between accessible categories is accessible).
A reflective subcategory is always closed under limits which exist in the ambient category (because the full inclusion is monadic, as noted above), and inherits colimits from the larger category by application of the reflector.
A morphism in a reflective subcategory is monic iff it is monic in the ambient category. A reflective subcategory of a well-powered category is well-powered.
See for instance (Borceux, vol 2, cor. 4.2.4) and see at idempotent monad – Properties – Algebras for an idempotent monad and localization.
Both the weak and strong versions of Vopěnka's principle are equivalent to fairly simple statements concerning reflective subcategories of locally presentable categories:
The weak Vopěnka's principle is equivalent to the statement:
This is AdamekRosicky, theorem 6.28
The strong Vopěnka's principle is equivalent to:
(Remark after corollary 6.24 in Adamek-Rosicky book).
If is cartesian closed, and is a reflective subcategory, then the reflector preserves finite products if and only if is an exponential ideal (i.e. implies for any ). In particular, if preserves finite products, then is cartesian closed.
A subcategory of a category of presheaves which is both reflective and coreflective is itself a category of presheaves , and the inclusion is induced by a functor .
This is shown in (BashirVelebil).
Whenever is a full subcategory of , we can say that objects of are objects of with some extra property. But if is reflective in , then we can turn this around and (by thinking of the left adjoint as a forgetful functor) think of objects of as objects of with (if we're lucky) some extra structure or (in any case) some extra stuff.
This can always be made to work by brute force, but sometimes there is something insightful about it. For example, a metric space is a complete metric space equipped with a dense subset. Or, an integral domain is a field equipped with numerator and denominator functions.
Complete metric spaces are mono-reflective in metric spaces; the reflector is called completion.
The category of sheaves on a site is a reflective subcategory of the category of presheaves on ; the reflector is called sheafification. In fact, categories of sheaves are precisely those accessible reflective subcategories, def. 3, of presheaf categories for which the reflector is left exact. This makes the inclusion functor precisely a geometric inclusion of toposes.
The non-full inclusion of unital rings into non-unital rings has a left adjoint (with monic units), whose reflector formally adjoins an identity element. However, we do not call it a reflective subcategory, because the “inclusion” is not full; see remark 1.
Notice that for a ring with unit, its reflection in the above example is not in general isomorphic to , but is much larger. But an object in a reflective subcategory is necessarily isomorphic to its image under the reflector only if the reflective subcategory is full. While the inclusion ‘ does have a left adjoint (as any forgetful functor between varieties of algebras, by the adjoint lifting theorem), this inclusion is not full (an arrow in ’ need not preserve the identity).
adjoint cylinder, describing the situation when the reflector has a further left adjoint
The relation of exponential ideals to reflective subcategories is discussed in section A4.3.1 of
Reflective and coreflective subcategories of presheaf categories are discussed in
Related discussion of reflective sub-(∞,1)-categories is in
The example of affine schemes in noncommutative algebraic geometry is in