nLab
extension

This entry is about extension of moprhisms, 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: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 p:YBp: Y\to B along an epimorphism (or more general map) f:XBf:X\to B to become 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 XX by an object YY is any object ZZ fitting in a short exact sequence

XiZpY X \stackrel{i}\hookrightarrow Z\stackrel{p}\rightarrowtail Y

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 XYX\coprod Y, then one looks for an additional structure in CC needed to equip the coproduct XYX\coprod Y 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

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 1XZY11\to X\to Z\to Y\to 1 corresponding to a group extension.

Revised on January 20, 2014 16:03:29 by Urs Schreiber (82.113.121.232)