nLab inclusion function

Given a set $X$ and a subset $A$ of $X$, the inclusion function of $A$ in $X$ is the function $i_A\colon A \to X$ given by

$$i_A(n) = n$$

for every element $n$ of $X$ that belongs to $A$.

Every inclusion function is an injection, and every injection is isomorphic (in the slice category $Set/X$) to an inclusion function. In appropriate categories, it is common to analogously interpret monomorphisms or regular monomorphisms as inclusions of subobjects. One then speaks of inclusion morphisms.

The inclusion function of $A$ in $X$ is the restriction to $A$ of the identity function on $X$. Conversely, the restriction to $A$ of any function $f\colon X \to Y$ is the composite of $f$ after the inclusion function:

$$i_A = {{\id_X}|_A} ,$$

$${{f}|_A} = f \circ i_A .$$