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 semantical extension and context extension.

Idea

Extension of morphisms

An extension

• of a morphism $f: A\to Y$

• along a morphism $i \colon A\hookrightarrow X$

  (often assumed to be a monomorphism)

is

• a morphism $\tilde{f} \colon X\to Y$

• such that $\tilde{f}\circ i = f$,

hence a completion of the span to a commuting diagram like this:

The dual problem is that of lifting a morphism $f \colon Y\to B$ through an (epi)morphism $p \colon X\to B$, 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 $C$.

Extension of an object as an enlargement

In something like the dual sense of the above, extension is also used to indicate a kind of enlargement of an object in the sense that the object being extended can be viewed as a subobject of the extension. This is the case for field extensions, $k \hookrightarrow K$.

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

Extension of representations

See at extended representation,

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)