The extension of a morphism$f: A\to Y$ along a monomorphism $i: A\hookrightarrow X$ is a morphism $\tilde{f}:X\to Y$ such that $\tilde{f}\circ i = f$. One sometimes, extends along more general morphisms than monomorphisms.

The dual problem is the problem of lifting a morphism $f: Y\to B$ through an epimorphism (or more general map) $p:X\to B$, giving a morphism $\tilde{f}: Y\to X$ such that $f = p\circ\tilde{f}$.

Classification of extensions in many categories is obtained using a forgetful functor$C\to D$ to a simpler category $D$, which preserves short exact sequences. For example, if all extensions in $D$ are isomorphic to $K\coprod Q$, then one looks for an additional structure in $C$ needed to equip the coproduct $K \coprod Q$ with a structure of an object in $C$ such that the $i$ and $p$ are morphisms in $C$ making above a short exact sequence in $C$.

The Tietze extension theorem is about extensions of continuous maps from a subspace to a normal toplogical space.

Group extensions

For example, in the category Grp of (possibly nonabelian) groups one has a short exact sequence usually denoted $1\to X\to Z\to Y\to 1$ corresponding to a group extension.