## § 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 $BG^\delta$.