Reflective subcategory

A full subcategory $\mathcal{D}\subseteq \mathcal{C}$ is reflective when its inclusion has a left adjoint $L:\mathcal{C}\to \mathcal{D}$, called the reflector. Equivalently, every object $A$ has a universal morphism $A\to LA$ into an object of $\mathcal{D}$.

The inclusion of compact Hausdorff spaces in topological spaces has a left adjoint. Its unit $X\to RX$ is universal among continuous functions from $X$ to compact Hausdorff spaces and has dense image. Completeness follows from products in a category and closed equalizers. For any map $X\to K$, its closed image has a dense subset of cardinality at most $|X|$ and hence cardinality at most $2^{2^{|X|}}$. Representatives of all such compact Hausdorff spaces and maps establish the solution-set condition, so the general adjoint functor theorem gives the reflection. The unit need not be injective or a topological embedding for an arbitrary $X$.

If a full subcategory is reflective in a complete category, it is complete. For an ambient limit $L$ of a diagram of reflected objects, the reflection unit $L\to JKL$ is invertible by the hom-set characterization of reflected objects. Thus $KL$ is an internal categorical limit. If the subcategory is replete, $L$ itself belongs to it. The reflector need not preserve arbitrary limits.

A reflective subcategory is left-exact when its reflector preserves finite limits. The fixed objects of a left-exact reflector are closed under finite limits.

For a left-exact reflective subcategory of a Cartesian closed category, an ambient exponential $B^A$ is already fixed by the reflector when $A,B$ are fixed. Indeed maps into $B^A$ transpose to maps from $X\times A$ into $B$, and reflection gives $L(X\times A)\cong LX\times A$. Precomposition with the reflection unit is therefore bijective for maps into $B^A$, characterizing a fixed object.

The reflector of a reflective subcategory is the left adjoint $L$ to its inclusion. Its unit ${\eta }_A:A\to LA$ is universal among morphisms from $A$ to objects of the subcategory.

For a subobject $m:A^{\prime }\hookrightarrow A$ and a left-exact reflector $L$ with unit $\eta$, define $c_A(A^{\prime })$ by the pullback in a categoryThis operation is monotone, inflationary, idempotent, and stable under pullback. If $A$ is fixed by $L$, then $A^{\prime }$ is fixed by $L$ exactly when $A^{\prime }$ is closed.