# nLab extension

category theory

## Applications

This entry is about extension of morphisms, dual to lift. For extensions in the sense of “added structure” see at group extension, Lie algebra extension, infinitesimal extension etc. In foundations and formal logic there is also extension (semantics) and context extension.

# Contents

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

### 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 $Q$ by an object $K$ is any object $X$ fitting in a short exact sequence

$K \stackrel{i}\to X \stackrel{p}\to Q$

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 $C$.

## 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.

Revised on May 23, 2017 14:02:21 by Urs Schreiber (92.218.150.85)