Reflective subcategory

Remarks

Equivalently, every object $c$ of $\mathcal{C}$ has a morphism $\eta_c \colon c \to R c$ with $R c$ in $\mathcal{B}$. Every morphism $c \to b$ with $b$ in $\mathcal{B}$ must factor through $\eta_c$ in exactly one way. Some authors also ask $\mathcal{B}$ to be closed under isomorphic objects. The dual notion is a coreflective subcategory.

Example: abelian groups form a reflective subcategory of groups, with reflector $R(H) = H/[H, H]$ and $\eta_H$ the quotient map. Every homomorphism $H \to A$ with $A$ abelian kills all commutators $x y x^{-1} y^{-1}$, so it factors uniquely through $H/[H, H]$.

Non-example: finite groups do not form a reflective subcategory of groups. A reflective subcategory closed under isomorphic objects contains every limit, formed in $\mathcal{C}$, of a diagram in $\mathcal{B}$. But the product of countably many copies of $\Z/2\Z$ is an infinite group.