Intuition about coreflective subcategories.

Question

When we have an inclusion of a full subcategory $I:B\hookrightarrow C$ which has a left adjoint we call it a reflector and say we have a reflective subcategory. An example is the inclusion of the category of complete metric spaces into metric spaces, where the reflector is given by the completion, another one is the inclusion of compact Hausdorff spaces in topological spaces, where the reflector is the Stone-Cech compactification. These examples build an intuition of a reflective subcategory as one that contains 'nice' objects , which can be built canonically for each object in the ambient category. Moreover such construction is idempotent.

Now, the dual notion of coreflective subcategory is that of an inclusion of a full subcategory $I:B\hookrightarrow C$ which has a right adjoint we call it a coreflector. And honestly, I have not found much more explanation.

How can we intuitively deal with this concept? Does it somehow express a concept dual to completion?

Answer 1

Your intuitive description of reflective subcategories doesn't distinguish left adjoints from right adjoints so you could apply it equally well to either. Anyway, I think the best way to get a handle on these sorts of things is through a bunch of examples.

Examples of reflective subcategories:

• The subcategory of abelian groups in the category of groups, with reflector the abelianization $G \mapsto G/[G, G]$. Note that it doesn't really make sense to think of this as a "completion." This is one of my preferred "prototypical" examples.
• The subcategory of torsion-free abelian groups in the category of abelian groups, with reflector the torsion-free quotient $A \mapsto A/A_{tors}$.
• The subcategory of fields in the category of integral domains and injections, with reflector the fraction field $D \mapsto \text{Frac}(D)$.
• Let $f : R \to S$ be an epimorphism of commutative rings (for example, a quotient or a localization). Then the category of $S$-modules is a reflective subcategory of the category of $R$-modules, with reflector the extension of scalars $M \mapsto M \otimes_R S$ (for example, a localization). Taking $R = \mathbb{Z}, S = \mathbb{Q}$ produces an example similar to the second one: the subcategory of torsion-free divisible abelian groups in the category of abelian groups, with reflector $A \mapsto A \otimes \mathbb{Q}$ (which has kernel $A_{tors}$).

Examples of coreflective subcategories:

• The subcategory of torsion abelian groups in the category of abelian groups, with coreflector the torsion subgroup $A \mapsto A_{tors}$. This example has the nice feature of looking quite dual to the $A \mapsto A/A_{tors}$ example; I think it's worth chewing on in detail.
• More generally, let $R$ be a commutative ring and let $S$ be a multiplicative subset of it. Call an element $m \in M$ of an $R$-module $S$-torsion if it is annihilated by some element of $S$; equivalently, if it lies in the kernel of the localization map $M \mapsto S^{-1} M \cong M \otimes_R S^{-1} R$. Then the subcategory of $S$-torsion modules is coreflective in the category of $R$-modules, with coreflector the $S$-torsion submodule.
• The subcategory of divisible abelian groups in the category of abelian groups, with coreflector the divisible subgroup $A \mapsto \{ a \in A : \forall n \in \mathbb{N}, \exists b \in A : a = nb \}$.
• The subcategory of groups in the category of monoids, with coreflector the group of units $M \mapsto M^{\times}$.
• The subcategory of connected topological groups in the category of topological groups, with coreflector taking the connected component of the identity $G \mapsto G_0$. If we restrict our attention to compact Hausdorff abelian groups then this example is related to the example of torsion-free groups by Pontryagin duality: the opposite of the category of abelian groups is equivalent to the category of compact Hausdorff abelian groups and the subcategory of torsion-free abelian groups (which are equivalently the filtered colimits of copies of $\mathbb{Z}^n$) is sent to the subcategory of connected compact Hausdorff abelian groups (which are equivalently the cofiltered limits of copies of $T^n$).

These algebraic examples work loosely as follows. A reflective subcategory consists of "nice" objects such that every object has a maximal "nice quotient" (with reflector given by this quotient), and a coreflective subcategory consists of "nice" objects such that every object has a maximal "nice subobject" (with coreflector given by this subobject). Probably there are examples a bit stranger than this (for example one could take the opposite category of metric spaces or topological spaces) but I think this is a decent place to start.