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.



Extension of morphisms

An extension of a morphism f:AYf: A\to Y along a monomorphism i:AXi: A\hookrightarrow X is a morphism f˜:XY\tilde{f}:X\to Y such that f˜i=f\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:YBf: Y\to B through an epimorphism (or more general map) p:XBp:X\to B, giving a morphism f˜:YX\tilde{f}: Y\to X such that f=pf˜f = p\circ\tilde{f}.

Extension of an object by another object

In a category CC with a notion of short exact sequence (e.g. any semiabelian category, Quillen exact category etc.) an extension of an object QQ by an object KK is any object XX fitting in a short exact sequence of the form

KiXpQ. 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 CDC\to D to a simpler category DD, which preserves short exact sequences. For example, if all extensions in DD are isomorphic to KQK\coprod Q, then one looks for an additional structure in CC needed to equip the coproduct KQK \coprod Q with a structure of an object in CC such that the ii and pp are morphisms in CC making above a short exact sequence in CC.

Other notions of extension


Extension of functions

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

extension theoremscontinuous functionssmooth functions
plain functionsTietze extension theoremWhitney extension theorem
equivariant functionsequivariant Tietze extension theorem

Group extensions

For example, in the category Grp of (possibly nonabelian) groups one has a short exact sequence usually denoted 1XZY11\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:

Last revised on November 18, 2020 at 11:28:38. See the history of this page for a list of all contributions to it.