- 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 has a left adjoint .
- In this case, usually the inclusion has more structure , and we the reflector 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 builds the completion.
- Eg 2: The subcategory of sheaves embeds into the category of presheaves. The reflector is sheafification.
§ General Contrast
- (the left adjoint to ) adds more structure. Eg: completion, sheafification.
- This is sensible because it's the left adjoint, so is kind of "free".
- (the right adjoint to ) 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 is a topological group. Consider the category of sets, call it .
- If we remove the topology on to become the discrete topology, we get a group called . This has a category of sets, called .