A contravariant functor, is like a covariant functor, except that it " turns morphisms around " (" reverses all the arrows ").

More specifically, every morphism in C must be assigned to a morphism in D. In other words, a contravariant functor acts as a covariant functor from the opposite category C < sup > op </ sup > to D.

Note that one can also define a contravariant functor as a covariant functor on the dual category.

The Hom functor is a natural example ; it is contravariant in one argument, covariant in the other.

Diagram: For categories C and J, a diagram of type J in C is a covariant functor.

Similar remarks apply to the colimit functor ( which is covariant ).

Tangent and cotangent bundles: The map which sends every differentiable manifold to its tangent bundle and every smooth map to its derivative is a covariant functor from the category of differentiable manifolds to the category of vector bundles.

Tensor products: If C denotes the category of vector spaces over a fixed field, with linear maps as morphisms, then the tensor product defines a functor C × C → C which is covariant in both arguments.

This is a functor which is contravariant in the first and covariant in the second argument, i. e. it is a functor Ab < sup > op </ sup > × Ab → Ab ( where Ab denotes the category of abelian groups with group homomorphisms ).

This defines a functor to Set which is contravariant in the first argument and covariant in the second, i. e. it is a functor C < sup > op </ sup > × C → Set.

is a covariant functor.

Mapping each object A in C to its associated hom-functor h < sup > A </ sup > = Hom ( A ,–) and each morphism f: B → A to the corresponding natural transformation Hom ( f ,–) determines a contravariant functor h < sup >–</ sup > from C to Set < sup > C </ sup >, the functor category of all ( covariant ) functors from C to Set.

One can interpret h < sup >–</ sup > as a covariant functor:

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.

If X is an object of a locally small category C, then the assignment defines a covariant functor.

Here one starts with some covariant additive functor F and some module X.

Chain complexes form a category: A morphism from the chain complex ( d < sub > n </ sub >: A < sub > n </ sub > → A < sub > n-1 </ sub >) to the chain complex ( e < sub > n </ sub >: B < sub > n </ sub > → B < sub > n-1 </ sub >) is a sequence of homomorphisms f < sub > n </ sub >: A < sub > n </ sub > → B < sub > n </ sub > such that for all n. The n-th homology H < sub > n </ sub > can be viewed as a covariant functor from the category of chain complexes to the category of abelian groups ( or modules ).

If F and G are ( covariant ) functors between the categories C and D, then a natural transformation η from F to G associates to every object X in C a morphism in D such that for every morphism in C, we have ; this means that the following diagram is commutative:

These are characterized by the assertion that the covariant derivative, which is the sum of the exterior derivative operator d and the connection A, transforms in the adjoint representation of the gauge group G. The square of the covariant derivative with itself can be interpreted as a g-valued 2-form F called the curvature form or field strength.

Suppose we are given a covariant left exact functor F: A → B between two abelian categories A and B.

The right derived functors of the covariant left-exact functor F: A → B are then defined as follows.

If F is a covariant functor, we say that F is

The most important examples of left exact functors are the Hom functors: if A is an abelian category and A is an object of A, then F < sub > A </ sub >( X ) = Hom < sub > A </ sub >( A, X ) defines a covariant left-exact functor from A to the category Ab of abelian groups.

Suppose that F is a ( 1-dimensional ) formal group law over R. Its formal group ring ( also called its hyperalgebra or its covariant bialgebra ) is a cocommutative Hopf algebra H constructed as follows.

Since what we now have here is a SO ( p, q ) gauge theory, the curvature F defined as is pointwise gauge covariant.

An alternate notation writes the connection form A as ω, the curvature form F as Ω, the canonical vector-valued 1-form e as θ, and the exterior covariant derivative as D.

Together with the preceding remark, it gives a criterion for a ( covariant ) functor F: C → D between triangulated categories satisfying certain technical conditions to have a right adjoint functor.

The notion of a directional derivative of a function from multivariable calculus is extended in Riemannian geometry to the notion of a covariant derivative of a tensor.

The current version of the standard from June 2006 contains some inconsistencies ( e. g. covariant redefinitions ).

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

Doing these constructions pointwise gives covariant and contravariant functors from the category of pointed differentiable manifolds to the category of real vector spaces.

It turns out that such inconsistencies arise from relativistic wavefunctions having a probabilistic interpretation in position space, as probability conservation is not a relativistically covariant concept.

The impetus arose from the fact that complete analytical solutions for the metric of a covariant N-body system have proven elusive in General Relativity.

The ( covariant ) hom-functor h < sup > A </ sup > sends X to the set of morphisms Hom ( A, X ) and sends a morphism f from X to Y to the morphism f o – ( composition with f on the left ) that sends a morphism g in Hom ( A, X ) to the morphism f o g in Hom ( A, Y ).

It is also possible that a rule which defines instantaneousness is contingent, by emerging dynamically from relativistic covariant laws combined with particular initial conditions.

If the chain complex depends on the object X in a covariant manner ( meaning that any morphism X → Y induces a morphism from the chain complex of X to the chain complex of Y ), then the H < sub > n </ sub > are covariant functors from the category that X belongs to into the category of abelian groups ( or modules ).

There is a covariant functor from the category of abelian groups to the category of torsion groups that sends every group to its torsion subgroup and every homomorphism to its restriction to the torsion subgroup.

There is another covariant functor from the category of abelian groups to the category of torsion-free groups that sends every group to its quotient by its torsion subgroup, and sends every homomorphism to the obvious induced homomorphism ( which is easily seen to be well-defined ).

In mathematics, a connector is a map which can be defined for a linear connection and used to define the covariant derivative on a vector bundle from the linear connection.

The covariant or spinfoam version of the dynamics developed during several decades, and crystalized in 2008, from the joint work of research groups in France, Canada, UK, Poland, and Germany, leading to the definition of a family of transition amplitudes, which in the classical limit can be shown to be related to a family of truncations of general relativity.

In category theory, there are covariant functors and contravariant functors.

* If C is a small category, then the functor category Set < sup > C </ sup > consisting of all covariant functors from C into the category of sets, with natural transformations as morphisms, is cartesian closed.