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.
The extension of a morphism along a monomorphism is a morphism such that . One sometimes, extends along more general morphisms than monomorphisms.
The dual problem is the problem of lifting a morphism along an epimorphism (or more general map) to become a morphism such that .
Classification of extensions in many categories is obtained using a forgetful functor to a simpler category , which preserves short exact sequences. For example, if all extensions in are isomorphic to , then one looks for an additional structure in needed to equip the coproduct with a structure of an object in such that the and are morphisms in making above a short exact sequence in .
The Tietze extension theorem is about extensions of continuous maps from a subspace to a normal toplogical space.