Contravariant in a sentence

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.

A pair of adjoint functors between two partially ordered sets is called a Galois connection (or, if it is contravariant, an antitone Galois connection).

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).

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.

If the transformation matrix of an index is the inverse matrix of the basis transformation, then the index is called contravariant and is traditionally denoted with an upper index (superscript).

In this way, Spec can be seen as a contravariant functor from the category of commutative rings to the category of topological spaces.

It allows the embedding of any category into a category of functors (contravariant set-valued functors) defined on that category.

Likewise, cotangent space is a contravariant functor, essentially the composition of the tangent space with the dual space above.

Ordinary functors are also called covariant functors in order to distinguish them from contravariant ones.

That is, instead of saying is a contravariant functor, they simply write (or sometimes ) and call it a functor.

The components of a more general tensor transform by some combination of covariant and contravariant transformations, with one transformation law for each index.

The coordinate independence of a tensor then takes the form of a covariant and/or contravariant transformation law that relates the array computed in one coordinate system to that computed in another one.

The functor Spec yields a contravariant equivalence between the category of commutative rings and the category of affine schemes; each of these categories is often thought of as the opposite category of the other.

The mnemonic "falling into something" can be helpful in remembering that "h A " is the contravariant hom-functor.

The operation is achieved by summing components for which one specified contravariant index is the same as one specified covariant index to produce a new component.

The proof in the contravariant case is completely analogous.

The vector is called covariant or contravariant depending on how the transformation of the vector's components is related to the transformation of the basis.

The vectors described in this article are a very special case of this general definition because they are contravariant with respect to the ambient space.

This follows, in part, from the fact the covariant Hom functor Hom(N, ) : C → Set preserves all limits in C. By duality, the contravariant Hom functor must take colimits to limits.