nLab
subfunctor

Definition

A subfunctor is a subobject in a functor category.

A subfunctor of a functor $G:C\to D$ between categories $C$ and $D$ is a pair $(F,i)$ where $F:C\to D$ is a functor and $i:F\to G$ is a natural transformation such that its components $i_M:F(M)\to G(M)$ are monic.

In fact one often by a subfunctor means just an equivalence class of such monic natural transformations; compare subobject.

A subfunctor is also called a subpresheaf . A subfunctor of a representable functor $\mathrm{Hom}(-,x)$ is precisely a sieve over the representing object $x$.

Properties

In a concrete category with images one can choose a representative of a subfunctor where the components of $i$ are genuine inclusions of the underlying sets; then a subfunctor is just a natural transformation whose components are inclusions. The naturality in terms of concrete inclusions just says that for all $f:c\to d$, $F(f)=G(f){\mid }_{F(c)}$. If the set-theoretic circumstances allow consideration of a category of functors, then a subfunctor is a subobject in such a category.

A subfunctor $(F,i)$ of the identity ${\mathrm{id}}_C:C\to C$ in a category with images is an often used case: it amounts to a natural assignment $c\mapsto F(c)\stackrel{i}{\hookrightarrow }c$ of a subobject to each object $c$ in $C$. For concrete categories with images then $F(f)=f{\mid }_{F(c)}$.