nLab
extension

Idea

Extension of morphisms

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 $p:Y\to B$ along an epimorphism (or more general map) $f:X\to B$ to become a morphism $\tilde{f}:Y\to X$ such that $f=p\circ \tilde{f}$.

Extension of an object by another object

In a category $C$ with a notion of short exact sequence (e.g. any semiabelian category, Quillen exact category etc.) an extension of an object $X$ by an object $Y$ is any object $Z$ fitting in a short exact sequence

$$X\stackrel{i}{\hookrightarrow }Z\stackrel{p}{\rightarrowtail }Y$$X \stackrel{i}\hookrightarrow Z\stackrel{p}\rightarrowtail Y

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 $X\coprod Y$, then one looks for an additional structure in $C$ needed to equip the coproduct $X\coprod Y$ 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 $C$.

Other notions of extension

• extension (semantics)

• context extension

Examples

Extension of functions

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.