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.

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 $f = p\circ\tilde{f}$.

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 $C$.

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 |

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.

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)

Last revised on November 18, 2020 at 06:28:38. See the history of this page for a list of all contributions to it.