contravariant
Speech parts
1. contravariant - Noun
2. contravariant - Adjective
Meaning
(category theory, of a functor) which reverses composition
(object-oriented programming) Using or relating to contravariance.
(linear algebra, of a vector) Scaling inversely with a change of basis.
contravariant (plural contravariants)
(algebra) A bihomogeneous polynomial in dual variables of x, y, ... and the coefficients of some homogeneous form in x, y, ... that is invariant under some group of linear transformations.
Examples
A pair of adjoint functors between two partially ordered sets is called a Galois connection (or, if it is contravariant, an antitone Galois connection).
Likewise, cotangent space is a contravariant functor, essentially the composition of the tangent space with the dual space above.
For contravariant functors the corresponding notions would be a functor that takes colimits to limits, or one that takes limits to colimits.
For the individual matrix entries, this transformation law has the form so the tensor corresponding to the matrix of a linear operator has one covariant and one contravariant index: it is of type (1,1).
Given an arbitrary contravariant functor G from C to Set, Yoneda's lemma asserts that : Naming conventions The use of "h A " for the covariant hom-functor and "h A " for the contravariant hom-functor is not completely standard.
However, the later book The geometry of schemes / David Eisenbud, Joe Harris (1998) reverses this and uses "h A " to mean the contravariant hom-functor.