Retraction

A retraction is a public statement made about an earlier statement that withdraws, cancels, refutes, or reverses the original statement or ceases and desists from publishing the original statement. The retraction may be initiated by the editors of a journal, or by the author(s) of the papers (or their institution). Retractions may or may not be accompanied by the author's further explanation as to how the original statement came to be made and/or what subsequent events, discoveries, or experiences led to the subsequent retraction. They are also in some cases accompanied by apologies for previous error and/or expressions of gratitude to persons who disclosed the error to the author.

Retractions always negate the author's previous public support for the original statement. Like original statements, retractions are in some cases incorrect. Retractions share with original statements the attribute that they are in some cases made insincerely, in some cases for personal gain, and in others under duress.

Section (category theory)

In category theory, a branch of mathematics, a section is a right inverse of some morphism. Dually, a retraction is a left inverse of some morphism. In other words, if f : X → Y and g : Y → X are morphisms whose composition f o g : Y → Y is the identity morphism on Y, then g is a section of f, and f is a retraction of g.

Every section is a monomorphism, and every retraction is an epimorphism.

In algebra the sections are also called split monomorphisms and the retractions split epimorphisms. In an abelian category, if f : X → Y is a split epimorphism with split monomorphism g : Y → X, then X is isomorphic to the direct sum of Y and the kernel of f.

Examples

In the category of sets, every monomorphism (injective function) with a non-empty domain is a section and every epimorphism (surjective function) is a retraction; the latter statement is equivalent to the axiom of choice.

In the category of vector spaces over a field K, every monomorphism and every epimorphism splits; this follows from the fact that linear maps can be uniquely defined by specifying their values on a basis.

Retraction (disambiguation)

A retraction is a public statement made to correct a previously made statement that was incorrect, invalid, or in error.

Retraction may also refer to