nLab
reflective subcategory

Definition

A full subcategory $C\subset D$ is reflective if its inclusion functor has a left adjoint

$$C\stackrel{\stackrel{T}{\leftarrow }}{\hookrightarrow }D.$$C \stackrel{\stackrel{T}{\leftarrow}}{\hookrightarrow} D \,.

The left adjoint is sometimes called the reflector.

The components of the unit

$$\begin{array}{cccc}& \nearrow & {\Downarrow }^{\eta }& {\searrow }^{\mathrm{Id}}\\ D& \stackrel{T}{\to }& C& \hookrightarrow & D\end{array}$$\array{ & \nearrow &\Downarrow^{\eta}& \searrow^{Id} \\ D &\stackrel{T}{\to}& C &\hookrightarrow & D }

of this adjunction “reflect” each object $d\in D$ into its image $Td$ in the reflective subcategory

$${\eta }_d:d\to Td.$$\eta_d : d \to T d \,.

This reflection is sometimes called a localization, although sometimes this term is reserved for the case when the functor $T$ is left exact.

Zoran: this is not universally accepted. In topos theory community yes. But in the setup of abelian categories, like categories of modules, people often use word localization even if left exactness is not met. If it is it is often said flat localization in those circles (though sometimes one says flat localization only if the stronger condition is satisfied: composed endofunctor is flat). The localization of the underlying ring (in the case of module categories) is the component of adjunction at that ring, and for Gabriel localizations (where $T$ is flat) the arget module is canonically a ring and the component of the adjunction is a ring morphism. But only if the localization is perfect this morphism of rings tell you all infrmation about the localization functor.

Mike Shulman: I changed it, how’s that?

If the reflector $T$ is faithful, the reflection is called a completion.

Characterizations

Given any adjoint pair $Q^{*}⊣Q_{*}$ of functors $Q^{*}:A\leftrightarrow B:Q_{*}$, the following are equivalent (Gabriel–Zisman):

• The right adjoint $Q_{*}$ is fully faithful.

• The counit $ϵ:Q_{*}{Q}^{*}\to 1_A$ of the adjunction is a natural isomorphism of functors.

• The monad $(Q^{*}{Q}_{*},Q^{*}\epsilon Q_{*},\eta )$ associated to the adjunction is idempotent.

• Let $S$ be the set of morphisms $s$ in $A$ such that $Q^{*}(s)$ is invertible in $B$; and $P_S:A\to A[S^{-1}]$ canonical localization functor; then the unique functor $H:A[S^{-1}]\to B$ such that $Q^{*}=H\circ P_S$ (given by the universal property of localization) is an equivalence of categories.

Zoran: Gabriel–Zisman neglect the set theoretical issues on the EXISTENCE of localizations. Is the last conditions really equivalent or we need to make some set-theoretical assumptions ?

Properties

A reflective subcategory is always closed under limits, and inherits colimits from the larger category by application of the reflector.

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’ 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 $\mathrm{Set}$.

Examples

• Complete metric spaces are mono-reflective in metric spaces; the reflector is called completion.

• The unital rings form a mono-reflective subcategory of possibly nonunital rings; the reflector formally adjoins an identity element.

• The category of sheaves on a site $S$ is a reflective subcategory of the category of presheaves on $S$; the reflector is called sheafification. In fact, categories of sheaves are precisely those reflective subcategories of presheaf categories for which the reflector is left exact. This makes the inclusion functor precisely a geometric morphism of topoi.

Property vs structure

Whenever $C$ is a full subcategory of $D$, we can say that objects of $C$ are objects of $D$ with some extra property. But if $C$ is reflective in $D$, then we can turn this around and (by thinking of the left adjoint as a forgetful functor) think of objects of $D$ as objects of $C$ 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, a possibly nonunital ring is a unital ring equipped with a unital homomorphism to the ring of integers.

Generalizations

• reflective (infinity,1)-subcategory