§ Coreflection

• A right adjoint to an inclusion functor is a coreflector.

§ Torsion Abelain Group -> Abelian Group

• If we consider the inclusion of abelian groups with torsion into the category of abelian groups, this is an inclusoin functor.
• This has right adjoint the functor that sends every abelian group into its torsion subgroup.
• See that this coreflector somehow extracts a subobject out of the larger object.

§ Group -> Monoid

• inclusion: send groups to monoids.
• coreflection: send monoid to its group of units. (extract subobject).

§ Contrast: Reflective subcategory

• To contrast, we say a category is reflective if the inclusion $i$ has a left adjoint $T$.
• In this case, usually the inclusion has more structure , and we the reflector $T$ manages to complete the larger category to shove it into the subcategory.
• Eg 1: The subcategory of complete metric spaces embeds into the category of metric spaces. The reflector $T$ builds the completion.
• Eg 2: The subcategory of sheaves embeds into the category of presheaves. The reflector is sheafification.

§ General Contrast

• $T$ (the left adjoint to $i$) adds more structure. Eg: completion, sheafification.
• This is sensible because it's the left adjoint, so is kind of "free".
• $R$ (the right adjoint to $i$) deletes structure / pulls out substructure. Eg: pulling out torsion subgroup, pulling out group of units.
• This is sensible because it's the right adjoint, and so is kind of "forgetful", in that it is choosing to forget some global data.

§ Example from Sheaves

• This came up in the context of group actions in Sheaves in geometry and logic.
• Suppose $G$ is a topological group. Consider the category of $G$ sets, call it $BG$.
• If we remove the topology on $G$ to become the discrete topology, we get a group called $G^\delta$. This has a category of $G^\delta$ sets, called $BG^\delta$.