nLab extension

This entry is about two senses of extension: extension of morphisms, dual to lift, and extension of objects (algebra extension). In foundations and formal logic there is also extension (semantics) and context extension.

Idea

Extension of morphisms

An 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 of the form

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

For further cases, such as group extension, Lie algebra extension, infinitesimal extension etc., see at algebra extension.

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 $C$.

Other notions of extension

• extension (semantics)

• context extension

• absolute extensor

• extension of distributions

Examples

Extension of functions

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

extension theorems	continuous functions	smooth functions
plain functions	Tietze extension theorem	Whitney extension theorem
equivariant functions	equivariant Tietze extension theorem

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.

Related concepts

• homotopy extension property

• Kan extension

• lifts and extensions

References

General discussion with an eye towards algebraic topology and the Tietze extension theorem:

• Norman Steenrod, Cohomology operations, and obstructions to extending continuous functions, Advances in Mathematics Volume 8, Issue 3, June 1972, Pages 371-416 (doi:10.1016/0001-8708(72)90004-7, pdf)