antiextension

Antiextension is a term used in several mathematical fields to denote a notion that is opposite or dual to the standard concept of extension. There is no single, universally accepted definition. In algebra and homological algebra, some authors use the term to refer to constructions that reverse the roles of the object being extended, effectively swapping the ends of a short exact sequence. For instance, while an extension of M by N is a short exact sequence 0 -> N -> E -> M -> 0, an antiextension could be described in informal terms as a construction that places M and N in swapped positions. However, in mainstream literature, the word coextension is more standard for this dual idea, and most references treat dual notions via opposite categories.

In category theory, antiextension may be used informally to describe phenomena in the opposite category or dualization of a universal property. In topology and analysis, the term has appeared occasionally to describe dual extension properties or obstruction phenomena, though again there is no standardized usage.

Because of the lack of a consistent definition, discussions of antiextension typically specify the exact context and the intended dual notion. When encountered, readers are advised to consult the source’s definitions to determine what the term means in that work.

See also: Extension, Coextension, Dual concepts, Opposite category.