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geometric or physical entities changes with a change of basis. Briefly, a contravariant vector is a list of numbers that transforms oppositely to a change of...

Look up covariance or contravariance in Wiktionary, the free dictionary. Covariance and contravariance may refer to: Covariance and contravariance of vectors...

"function from Cat to String" because the function type constructor is contravariant in the parameter type. Here, the subtyping relation of the simple types...

n t {\displaystyle \mathrm {Contravariant} \circ \mathrm {Covariant} \to \mathrm {Contravariant} } Note that contravariant functors reverse the direction...

embedding of any locally small category into a category of functors (contravariant set-valued functors) defined on that category. It also clarifies how...

in topos theory Y is required to be Ω. There is both a covariant and contravariant power set functor, P: Set → Set and P: Set op → Set. The covariant functor...

transform in the same way. The inverse of a covariant transformation is a contravariant transformation. Whenever a vector should be invariant under a change...

called a profunctor. A presheaf that is naturally isomorphic to the contravariant hom-functor Hom(–, A) for some object A of C is called a representable...

In higher category theory in mathematics, co- and contravariant model structures are special model structures on slice categories of the category of simplicial...

where the hat denotes the components in the new basis. This is called a contravariant transformation law, because the vector components transform by the inverse...

is equivalent to defining a contravariant vector to be a tensor of contravariant rank one. Alternatively, a contravariant vector is defined to be a tangent...

in homotopy theory gives necessary and sufficient conditions for a contravariant functor F on the homotopy category Hotc of pointed connected CW complexes...

In mathematics invariant theory, the bracket ring is the subring of the ring of polynomials k[x11,...,xdn] generated by the d-by-d minors of a generic...

to sources, and targets are mapped to targets (or, in the case of a contravariant functor, sources are mapped to targets and vice-versa). A third fundamental...

not to any contravariant indices which happen to be placed intermediately between the brackets. Similarly if brackets enclose contravariant indices – the...

a ≡ ∂ a ϕ {\displaystyle \phi _{;a}\equiv \partial _{a}\phi } For a contravariant vector field λ a {\displaystyle \lambda ^{a}} , we have: λ a ; b ≡ ∂...

tensor is a tensor which is neither strictly covariant nor strictly contravariant; at least one of the indices of a mixed tensor will be a subscript (covariant)...

_{j}^{i}} . Note the careful use of upper and lower indexes, to distinguish contravariant and covariant vectors. The pullback induces (defines) a metric tensor...

{b1, b2, b3} is the contravariant basis, and {b1, b2, b3} is the covariant (a.k.a. reciprocal) basis. The covariant and contravariant basis vectors types...

algebraic geometry, an inverse image functor is a contravariant construction of sheaves; here “contravariant” in the sense given a map f : X → Y {\displaystyle...

covariant functor and fixing the second argument naturally gives a contravariant functor. This is an artifact of the way in which one must compose the...

is a contravariant of degree 3 and class 3. The quippian Q of a ternary cubic is a contravariant of degree 5 and class 3. The Hermite contravariant Π is...

associates to a contravariant pseudofunctor a fibered category, and conversely, each fibered category is induced by some contravariant pseudofunctor. Because...

}}_{\mu }x^{\mu },} where lower and upper indices label covariant and contravariant components respectively, and the summation convention is applied. It...

mathematics, particularly algebraic topology, cohomotopy sets are particular contravariant functors from the category of pointed topological spaces and basepoint-preserving...

with a totally symmetric contravariant tensor, when every choice is equivalent, and with a totally antisymmetric contravariant tensor (a.k.a. a k-vector)...

covariant and contravariant refer to how objects transform under general coordinate transformations. Both covariant and contravariant four-vectors can...

general contraction is denoted by labeling one covariant index and one contravariant index with the same letter, summation over that index being implied...

topological spaces. Formally, a simplicial set may be defined as a contravariant functor from the simplex category to the category of sets. Simplicial...

{\frac {d}{d\mu }}p_{\lambda }-F_{\lambda }=0} can be transformed into a contravariant form d p k d μ + g k λ ∂ g λ i ∂ x j v j p i = F k , {\displaystyle...

Plasmas 1:3'(Roscoe White), Writing B → {\displaystyle {\vec {B}}} by contravariant basis ( ∇ Ψ , ∇ ϕ , ∇ ζ ) {\displaystyle (\nabla \Psi ,\nabla \phi ...

with pushforwards, “transfer” maps. Precisely, it is, by definition, a contravariant additive functor from the category of finite correspondences (defined...

contravariant functor from the category of (nice) spaces to the category of rings. A bivariant theory is a functor both covariant and contravariant;...

"co-vary" with vectors). In the case of orthogonal coordinates, the contravariant basis vectors are easy to find since they will be in the same direction...

Figure 6. Contravariant components of the stress–energy tensor...

by William Lawvere in 1986. That is a duality between covariant and contravariant representable presheaves associated with an objects of categories under...

In mathematics, a quippian is a degree 5 class 3 contravariant of a plane cubic introduced by Arthur Cayley (1857) and discussed by Igor Dolgachev (2012...

f_{*}} or f ∗ , {\displaystyle f^{*},} accordingly as it is covariant or contravariant. To indicate a map that goes "the wrong way" – a functor that has the...

components of contravariant vectors (vectors), lower indices represent components of covariant vectors (covectors). They transform contravariantly or covariantly...

pushforward describe the transformation properties of covariant and contravariant tensors used in more traditional (coordinate dependent) approaches to...

how one could imagine a covariant version of a contravariant vector. The inner product of two contravariant vectors could equally well be thought of as the...

} , that is a contravariant functor from Δ {\displaystyle \Delta } to another category. For instance, simplicial sets are contravariant with the codomain...

:\operatorname {Hom} (A,-)\rightarrow F} is a natural isomorphism. A contravariant functor G {\displaystyle G} from C {\displaystyle {\mathcal {C}}} to...

dimensions; similarly four-momentum is a four-vector in spacetime. The contravariant four-momentum of a particle with relativistic energy E and three-momentum...

polynomial in the coefficients of a form, a covariant variable, and a contravariant variable Concomitant (invariant theory), a relative invariant of GL(V)...

by a momentum-convective term. Most of this article works with the contravariant form, Tμν of the stress–energy tensor. However, it is often convenient...

covariant or contravariant depending on how the transformation of the vector's components is related to the transformation of coordinates. Contravariant vectors...

lattice is described by a set of unit cell, direct lattice basis vectors (contravariant or polar) called a, b, and c, or equivalently by the lattice parameters...

algebraic topology, a presheaf of spectra on a topological space X is a contravariant functor from the category of open subsets of X, where morphisms are...

the study of the polynomial invariants for an arbitrary number of (contravariant or covariant) variables for a standard classical group action; second...

introduced by Glen E. Bredon, is a type of equivariant cohomology that is a contravariant functor from the category of G {\displaystyle G} -complexes with equivariant...

represented by the repetition of an index label, where one label is contravariant (an upper index corresponding to the factor V {\displaystyle V} ) and...

In mathematics, a Γ-object of a pointed category C is a contravariant functor from Γ to C. The basic example is Segal's so-called Γ-space, which may be...

opposite along f {\displaystyle f} in the first component since it is contravariant. The composition of two dinatural transformations need not be dinatural...

× 4 matrices Λ. The action of a Lorentz transformation on a general contravariant four-vector X (like the examples above), regarded as a column vector...

object in the category of topological spaces. In other words, it is a contravariant functor from the simplex category Δ to the category of topological spaces...

Mathematically, direct and reciprocal lattice vectors represent covariant and contravariant vectors, respectively. The Brillouin zone is a Wigner–Seitz cell of...

In mathematical invariant theory, an evectant is a contravariant constructed from an invariant by acting on it with a differential operator called an...

interchanged. The index subset must generally either be all covariant or all contravariant. For example, T i j k … = − T j i k … = T j k i … = − T k j i … = T...

network of tensors. Sensory input = covariant tensor Motor output = contravariant tensor Cerebellar neuronal network = metric tensor that transforms the...

a pseudovector (or axial vector). More formally, a true vector is a contravariant vector, see: Covariance and contravariance of vectors. True vector display...

the Hom functor, which makes the twist come from the fact that it is contravariant in the first entry and covariant in the second entry. It can be generalized...

return new B(); } } More specifically, covariant (wide to narrower) or contravariant (narrow to wider) return type refers to a situation where the return...

the matrices of the action of a set of orthogonal basis vectors for contravariant vectors in Minkowski space, the column vectors on which the matrices...

In topology, a branch of mathematics, a cubical set is a set-valued contravariant functor on the category of (various) n-cubes. Cubical sets have been...

space. The components of this vector transform between themselves contravariantly under rotations in space. Similarly, a dual (or co-) vector field attaches...

normalized future-directed timelike tangent vector to a world line, and is a contravariant vector. Though it is a vector, addition of two four-velocities does...

mapping cone, the sequence: F(X) → F(Y) → F(C(f)) is exact. If F is a contravariant functor, it is half-exact if for each sequence of spaces as above, the...

the magnetic field, and c the speed of light. The four-current is the contravariant four-vector which combines electric charge density ρ and electric current...

:T_{1}\rightarrow T_{2}}} The parameter type of S1 → S2 is said to be contravariant because the subtyping relation is reversed for it, whereas the return...

} provide a link between covariant and contravariant components of other tensors. Contracting the contravariant index of a tensor with one of a covariant...

a timelike component ct and spacelike component x = (x, y, z), in a contravariant position four-vector with components: X ν = ( X 0 , X 1 , X 2 , X 3...

generalization of the Yoneda lemma to 2-categories. Precisely, given a contravariant pseudofunctor F {\displaystyle F} on a category C, it says: for each...

tensor density field in two different ways. It may be regarded as a contravariant tensor density of weight +1 or as a covariant tensor density of weight...

R to that of X. The use of the superscript is meant to indicate its contravariant nature: it reverses the direction of the map. For example, if X, Y are...

functor Hom(N, –) : C → Set preserves all limits in C. By duality, the contravariant Hom functor must take colimits to limits. If a diagram F : J → C has...

and delegates can have their type parameters marked as covariant or contravariant using keywords out and in respectively. These declarations are then...

vector U {\displaystyle \mathbf {U} } expressed this way transform as a contravariant tensor, as shown in the expression U = U i e i {\displaystyle \mathbf...

must satisfy, given that it is a presheaf, which is by definition a contravariant functor F : O ( X ) → C {\displaystyle {\mathcal {F}}:{\mathcal {O}}(X)\rightarrow...

copies of V∗ in our product, the tensor is said to be of type (m, n) and contravariant of order m and covariant of order n and of total order m + n. The tensors...

the case of C*-algebras with unit, the spectrum map gives rise to a contravariant functor from the category of commutative C*-algebras with unit and unit-preserving...

of M {\displaystyle M} (whose sections are called vector fields or contravariant vector fields in physics) and V ∗ = T ∗ M {\displaystyle V^{*}=T^{*}M}...

covariant and contravariant functors that assign to a map a group and a ring respectively. It generalizes a cohomology theory, which is a contravariant functor...

+,+), while a minus sign (-) is chosen for (+,-,-,-). The inverse (contravariant) optical metric tensor is g ^ a b = g a b ± ( 1 − ϵ μ ) u a u b , {\displaystyle...

global Spec of a quasi-coherent sheaf of algebras: this results in the contravariant functor Spec X {\displaystyle \operatorname {Spec} _{X}} from the category...

category of topological spaces) taking values in simplicial sets (i.e., a contravariant functor from the site to the category of simplicial sets). Equivalently...

is simply the category D C op {\displaystyle D^{C^{\text{op}}}} of contravariant functors from C {\displaystyle C} to D {\displaystyle D} with natural...

timelike components and 1, 2, 3 for spacelike components. There are n contravariant indices and m covariant indices. In special and general relativity,...

of contravariant rank one. In this more general framework, higher rank tensors can also have arbitrarily many and mixed covariant and contravariant ranks...

the density is a function in mixed contravariant-covariant (xi, pi) phase space as opposed to fully contravariant (xi, pi) phase space. In physical cosmology...

-)\simeq \operatorname {Hom} _{C}(-,G(-))} of bifunctors (which are contravariant in the first variable.) algebra for a monad Given a monad T in a category...

cospan K in a category C is a functor K : Λop → C; equivalently, a contravariant functor from Λ to C. That is, a diagram of type Λ op = ( − 1 → 0 ← +...

algebraic geometry, a functor represented by a scheme X is a set-valued contravariant functor on the category of schemes such that the value of the functor...

covariant indices, and transform via pullbacks, while upper indices are contravariant, transforming under pushforwards. Thus, the structure coefficients are...

=U_{0}V_{0}-U_{1}V_{1}-U_{2}V_{2}-U_{3}V_{3}\,.} In all the coordinate systems, the (contravariant) relativistic four-velocity is defined by U ≡ d R d τ = γ d R d t ,...

including derivatives. In an unrelated use, superscripts also indicate contravariant tensors in Ricci calculus. The charges of ions and subatomic particles...

(D)} denotes the category of contravariant functors from D {\displaystyle D} to the category of sets; such a contravariant functor is frequently called...

A[u]. As vectors change with the inverse of B (vectors coordinates are contravariant) its inverse transformation is [v] = B[v′]. Substituting this in the...

{\displaystyle Z_{ij}} and Z i j {\displaystyle Z^{ij}} are covariant and contravariant metric tensors, δ j i {\displaystyle \delta _{j}^{i}} is the Kronecker...

representable functors one can state the above result as follows. The contravariant functor, which associates to each k-variety T the set of families of...

length contraction and time dilation and some relations to covariant and contravariant vectors were demonstrated by them. Gruner extended this method in subsequent...

components of the two-vector. Notice that both indices of the components are contravariant. This is always the case for two-vectors, by definition. A bivector...

polymorphism. In some languages subtypes may also possess covariant or contravariant return types and argument types respectively. Certain languages, for...

applications, it is important to generalize the construction to even a contravariant pseudofunctor F {\displaystyle F} (the covariant case is similar). Namely...

locally finitely presentable category. If C is an arbitrary category, the contravariant functors from C to Set are often an important object of study. If A...

distinct from the notion of a topological half-exact functor. If G is a contravariant additive functor from P to Q, we similarly define G to be exact if whenever...

{\frac {\mathrm {d} \xi ^{a}}{\mathrm {d} t}},} where Fa is the a-th contravariant component of the resultant force acting on the particle, Γabc are the...

also start with a contravariant left-exact functor F {\displaystyle F} ; the resulting right-derived functors are then also contravariant. The short exact...

{\displaystyle R^{op}} -module, or that of an opposite category so that a contravariant functor C → D {\displaystyle C\to D} can be regarded as an ordinary...

notation. Formulae are given in SI units and Gaussian-cgs units. The contravariant electromagnetic four-potential can be defined as: in which ϕ is the...

"coefficient field" K of characteristic zero. A Weil cohomology theory is a contravariant functor H ∗ : { smooth projective varieties over  k } ⟶ { graded  K...

structural mechanics software. It can be shown that the stress tensor is a contravariant second order tensor, which is a statement of how it transforms under...

the context of schemes. Then a presheaf on X {\displaystyle X} is a contravariant functor from O ( X ) {\displaystyle O(X)} to the category of sets, and...

of smooth quasi-projective schemes Sm over a field k consists of a contravariant functor A* from Sm to commutative graded rings, together with push-forward...

a fixed R {\displaystyle R} -module B {\displaystyle B} . This is a contravariant functor, which can be viewed as a left exact functor from the opposite...

_{3})=\cos(\phi )} which are quantities that will be useful later on. The contravariant basis is given by e 1 = e 2 × e 3 g = e 2 × e 3 cos ⁡ ( ϕ ) {\displaystyle...

classes, multiple inheritance, anonymous functions, operator overloading, contravariant type system. The original Berkeley implementation (last stable version...

is a contravariant functor from the category of open subsets to sets. By analogy we define an étale presheaf on a scheme X to be a contravariant functor...

iteration A data type in method overriding that is neither covariant nor contravariant Class invariant, an invariant used to constrain objects of a class Invariant...

elements of the respective modules. Analogous formulas also hold for contravariant tensors, as well as tensors of mixed variance. Although in many cases...

algebra. The terminology tends to hide the fact that cohomology, a contravariant theory, is more natural than homology in many applications. At a basic...

while flat maps are contravariant. Also, whenever Y {\displaystyle Y} is smooth, any map to Y {\displaystyle Y} is contravariant. If E → X {\displaystyle...

is a morphism of frames. If we define Ω(f) = f −1 then Ω becomes a contravariant functor from the category Top to the category Frm of frames and frame...

spaces one obtains a ring homomorphism on cohomology rings, which is contravariant. Specifically, given a sequence of cohomology groups Hk(X;R) on X with...

one row index and one column index, also called covariant order 1 and contravariant order 1; see Tensor (intrinsic definition) for details. The tensor rank...

the fields in the new frame will be given by the new components. In contravariant matrix form with metric signature (+,−,−,−), F μ ν = [ 0 − E x / c −...

{\displaystyle \delta _{j}^{i}} with a covariant index j {\displaystyle j} and contravariant index i {\displaystyle i} : δ j i = { 0 ( i ≠ j ) , 1 ( i = j ) . {\displaystyle...

G-bundles over X {\displaystyle X} . This b G {\displaystyle b_{G}} is a contravariant functor from Top (the category of topological spaces and continuous...

category of abelian groups A b {\displaystyle Ab} . First we construct a contravariant embedding H : A → L {\displaystyle H:{\mathcal {A}}\to {\mathcal {L}}}...

{x} /\partial x^{i}} refer to the unnormalized local covariant and contravariant bases respectively, g i j {\displaystyle g^{ij}} is the inverse metric...

from R to the category of abelian groups Ab, and right R-modules are contravariant additive functors. This suggests that, if C is any preadditive category...

covariants. The degree and class of a contravariant are its degrees in the two types of variable. Contravariants generalize invariants and are special...

\mathbf {P} =\left({\frac {E}{c}},\mathbf {p} \right)\,} (these are the contravariant components). The Minkowski inner product ⟨⋅, ⋅⟩ of this vector with...

category C {\displaystyle {\mathcal {C}}} equipped with an involutive contravariant endofunctor † {\displaystyle \dagger } which is the identity on objects...

Heisenberg group W(V) = F[H(V∗)]. Since passing to group algebras is a contravariant functor, the central extension map H(V) → V becomes an inclusion Sym(V)...

frame f is changed by a matrix A via The inverse metric transforms contravariantly, or with respect to the inverse of the change of basis matrix A. Whereas...

Hom G ⁡ ( − , M ) {\displaystyle \operatorname {Hom} _{G}(-,M)} is a contravariant functor and reverses the arrows, applying Hom G ⁡ ( − , M ) {\displaystyle...

0 depending on the presentation of the field: covariant, mixed, and contravariant respectively. This doesn't necessarily come symmetric directly from...

curved space, where g α β {\displaystyle g^{\alpha \beta }} are the contravariant coordinates of the metric tensor (i.e., the inverse metric) solved from...

categorically, this is not just an isomorphism of endomorphism algebras, but a contravariant equivalence of categories – see § Categorical considerations. A locally...

satisfy the transformation law Such a transformation law is called contravariant. A similar transformation law characterizes vector fields in physics:...

full theory. This includes proving fundamental properties, such as the contravariant functoriality and a projective bundle formula, as well as constructing...

{\displaystyle S\longmapsto DS} . Note that H k {\displaystyle H^{k}} is a contravariant functor while H n − k {\displaystyle H_{n-k}} is covariant. The family...

j_{*}:H_{\mathrm {c} }^{q}(U)\to H_{\mathrm {c} }^{q}(X)} . They also demonstrate contravariant behavior with respect to proper maps - that is, maps such that the inverse...

Group effort The cohomology concept emerges in algebraic topology, contravariant and dual to homology. In the setting of de Rham, cohomology gives classes...

factors it has. Sometimes these ranks are referred to as covariant and contravariant ranks, signifying tangent and cotangent ranks, respectively. A frame...

{\partial F_{\beta \gamma }}{\partial x^{\alpha }}}=0} where εδαβγ is the contravariant Levi-Civita symbol. Notice the cyclic permutation of indices in this...

left-exact functors, there is for example Proposition—Let F, G be left-exact contravariant functors from the category of modules over a commutative ring R to abelian...

abelian tensor category M M ( k ) {\displaystyle MM(k)} , together with a contravariant functor Var ⁡ ( k ) → M M ( k ) {\displaystyle \operatorname {Var} (k)\to...

vanitas, vanity, vaunt varius vari- varying bivariate, contravariance, contravariant, countervair, covariance, covariant, covariate, covariation, intervarietal...

invariants. Cayleyan A contravariant. characteristic See Sylvester (1853, Glossary p. 543–548) class The class of a contravariant or concomitant is its...

naturality of the isomorphism is also a consequence of the diagram. Thus the contravariant hom-functor changes coproducts into products. Stated another way, the...

geometry come in different kinds: for example embeddings of submanifolds. Contravariant objects (which is to say, objects that have pullbacks; these are called...

f} the inverse image functor f ∗ {\displaystyle f^{*}} is almost a contravariant functor from E {\displaystyle E} to the category of categories. However...

functorial on ( S c h / S ) {\displaystyle (\mathrm {Sch} /S)} forming a contravariant 2-functor ( R ( − ) , U ( − ) , s , t , m ) : ( S c h / S ) o p → Cat...

{\displaystyle F} be a contravariant functor from C {\displaystyle {\mathcal {C}}} to the category of sets Set. A contravariant functor G {\displaystyle...

particular kind of a group functor. A scheme may be thought of as a contravariant functor from the category S c h S {\displaystyle {\mathsf {Sch}}_{S}}...

\mathbf {b} _{k}} The components v k {\displaystyle v^{k}} are the contravariant components of the vector v {\displaystyle \mathbf {v} } . The reciprocal...

concretizable. The forgetful functor which arises in this way is the contravariant powerset functor Setop → Set. It follows from the previous example that...

over a ring R, is given by the tensor product over R, and Spec is a contravariant functor, the pullback of two affine schemes Spec(A) and Spec(B) over...

arrows i → j if and only if i ≤ j. An inverse system is then just a contravariant functor I → C. Let C I o p {\displaystyle C^{I^{\mathrm {op} }}} be...

called a direct system of objects and morphisms. If the diagram is contravariant then it is called an inverse system. A cone with vertex N of a diagram...

p_{x},p_{y},p_{z}\right),} τ the proper time of the particle, Fαβ the contravariant electromagnetic tensor F α β = ( 0 − E x / c − E y / c − E z / c E x...

meaning that its objects do not form a set. An étale presheaf on X is a contravariant functor from Ét(X) to the category of sets. A presheaf F is called an...

theory, ω X {\displaystyle \omega _{X}} is an object representing the contravariant functor F ↦ H n ⁡ ( X , F ) ∗ {\displaystyle F\mapsto \operatorname...

(Cayley–Klein model) Cayley–Klein metric Goldstein (1980) considers a passive (contravariant, or "alias") transformation, rather than the active (covariant, or "alibi")...

superscripts are used instead of subscripts to distinguish covariant from contravariant entities, see covariance and contravariance of vectors and raising and...

Vect ⁡ ( X ) {\displaystyle X\mapsto \operatorname {Vect} (X)} as a contravariant functor from Top to Set by sending a map ƒ:X → Y to the pullback ƒ*...

the same as a functor (the term is used to be in contrast to the term contravariant functor). Covariance and contravariance (computer science), a system...

nonlinearities.[citation needed] Note that the transition between "upper" ("contravariant") and "lower" ("covariant") vector or tensor components is trivial for...

space V then Proj S is P(V). This Proj construction gives rise to a contravariant functor from the category of graded commutative rings and surjective...

thought of as a special case of a tensor with one covariant, and one contravariant index, and thus two T {\displaystyle {\mathcal {T}}} 's are required...

cohomology groups Ω G ∗ ( X ) {\displaystyle \Omega _{G}^{*}(X)} are contravariant in X. The cobordism groups defined above are, from this point of view...

simplicial diagram. Formally, an augmented simplicial diagram is a contravariant functor from the augmented simplex category Δ aug {\displaystyle \Delta...

partially ordered set. Steiner demonstrated that the equivalence is a contravariant lattice isomorphism preserving arbitrary meets and joins as well as...

abelian variety. This association is a duality in the sense that it is contravariant functorial, i.e., it associates to all morphisms f : A → B {\displaystyle...

of finite groups Voigt notation Here, upper and lower indices denote contravariant and covariant components, respectively, though the distinction can be...

the regular maps. Regular maps between affine varieties correspond contravariantly in one-to-one to algebra homomorphisms between the coordinate rings:...

{\displaystyle -W} with contravariant indices. Similar statements apply when the two indices are contravariant or are mixed covariant and contravariant. If T α β {\displaystyle...

dual of vector spaces and the transpose of linear maps is therefore a contravariant functor from the category of vector spaces over F {\displaystyle F}...

Each groupoid object in a category C (if any) may be thought of as a contravariant functor from C to the category of groupoids. This way, each groupoid...

described nicely using the language of category theory. There is a contravariant functor T : T o p → S e t {\displaystyle T:\mathbf {Top} \to \mathbf...

that explicitly. Kronecker delta Levi-Civita symbol Covariant tensor Contravariant tensor The classical interpretation is by components. For example, in...

object Dual (category theory) Duality (mathematics) Adjoint functor Contravariant functor Opposite functor "Is there an introduction to probability theory...

written to and read from, implying that it is neither covariant nor contravariant, and is instead invariant. Therefore, Circle is not a subtype of Ellipse...

{x}}^{l}{\dot {x}}^{k}=F^{j}} , where F j {\displaystyle F^{j}} are the contravariant components of the force per unit mass, and Γ j l k {\displaystyle \Gamma...

using physical components (which are neither covariants ("column") nor contravariants ("row") ). However, if we chose a non-orthogonal curvilinear coordinate...

K^{n}} (respectively, K ~ n {\displaystyle {\widetilde {K}}^{n}} ) is a contravariant functor from the homotopy category of (pointed) spaces to the category...

^{i}\\-\sigma ^{i}&0\end{pmatrix}}\,} as well as the definition of contravariant four-momentum in natural units, p μ = ( E , p x , p y , p z ) {\displaystyle...

dualities between the module categories, where the functors used are contravariant rather than covariant. This theory, though similar in form, has significant...

Hom(X, G) from X to G such that the association of X to Hom(X, G) is a (contravariant) functor from C to the category of groups. Yet another way to state...

and delegates can have their type parameters marked as covariant or contravariant, using keywords out and in, respectively. These declarations are then...

Therefore, a tangent vector of a smooth curve will transform as a contravariant tensor of order one under a change of coordinates. Let f : R n → R {\displaystyle...

^{j0}X_{0}+\eta ^{ji}X_{i}=\delta ^{ji}X_{i}=X_{j}\,.} So the index-raised contravariant 4-position is: X μ = ( c t , x , y , z ) . {\displaystyle X^{\mu }=(ct...

morphism Tn(M '') → Tn-1(M'). The notions of contravariant cohomological δ-functor between A and B and contravariant homological δ-functor between A and B can...

conjectural) applications to the theory of motives. There is a covariant and a contravariant functoriality of the group of algebraic cycles. Let f : X → X' be a...

i.e. are covariant under a change of basis, while upper indices are contravariant. The structure constants obviously depend on the chosen basis. For Lie...

homomorphism makes sheaf cohomology with constant coefficients into a contravariant functor from topological spaces to abelian groups. For any spaces X...

diagram: If both F {\displaystyle F} and G {\displaystyle G} are instead contravariant functors, the vertical arrows in the right diagram are reversed. If...

f:M\rightarrow N} is a smooth map and Ω k {\displaystyle \Omega ^{k}} is the contravariant smooth functor that assigns to each manifold the space of k {\displaystyle...

given using the functor G(A)=HomR(A,B). For a fixed module B, this is a contravariant left exact functor, and thus we also have right derived functors RnG...

is defined to be a presheaf on Δ {\displaystyle \Delta } , i.e. a (contravariant) functor S : Δ op → Set {\displaystyle S:\Delta ^{\text{op}}\to {\text{Set}}}...

vector because of this, including Gibbs vectors, pseudovectors and contravariant vectors. The latter of these two, in plane-based GA, map to the concepts...

cases, and so is covariant (whereas a restricted representation is contravariant). Attempts to specify a direct construction have only produced some...

cyclic set is a contravariant functor from the cyclic category to sets. More generally a cyclic object in a category C is a contravariant functor from the...

displacement field D and the auxiliary magnetic field H form an antisymmetric contravariant rank-2 tensor density of weight +1. In vacuum, this is given by D μ...

position four-vector, the summation convention over the covariant and contravariant spacetime indices α and β is used, each index takes the value 0 for...

n C {\displaystyle \operatorname {Vect} _{n}^{\mathbb {C} }} be the contravariant functor that, to a CW complex X, assigns the set of isomorphism classes...

_{X}^{t}(Y){\bigr )},\quad \alpha \in T_{\Phi _{X}^{t}(p)}^{*}M,Y\in T_{p}M} For contravariant tensors, one extends the inverse ( T p Φ X t ) − 1 : T Φ X t ( p ) M...

reversed. Using the parlance of category theory, this amounts to a contravariant functor between two categories C and D: F: C → D which for any two objects...

Y)_{n}\to {\textbf {Hom}}(X,Y)_{m}} (since the first factor of Hom is contravariant) defined by sending a map f : X × Δ n → Y {\displaystyle f:X\times \Delta...

proper submodule of M(λ) The quotients M(λ)i/M(λ)i+1 have non-degenerate contravariant bilinear forms. The Jantzen sum formula holds: ∑ i > 0 Ch ( M ( λ )...

O(N) is orthogonal, no distinction will be made between covariant and contravariant indices. The Lagrangian density is given by L = 1 2 ∂ μ ϕ a ∂ μ ϕ a...

period T and inverse wavelength λ. When written out explicitly its contravariant and covariant forms are: K μ = ( ω c , k x , k y , k z ) K μ = ( ω c...

partially ordered sets is called a Galois connection (or, if it is contravariant, an antitone Galois connection). See that article for a number of examples:...

R {\displaystyle R} -homomorphism. Character module formation is a contravariant functor from the category of left R {\displaystyle R} -modules to the...

field). Assigning to each variety its function field yields a duality (contravariant equivalence) between the category of varieties over k {\displaystyle...

in the language of exterior algebra, and angular momentum becomes a contravariant second order antisymmetric tensor L = x ∧ p {\displaystyle \mathbf {L}...

homomorphism h from J(P) to J(Q). In category theoretic terminology, J is a contravariant hom-functor J = Hom(—,2) that defines a duality of categories between...

{\displaystyle f\circ h=g} . The following conditions are equivalent: The contravariant functor Hom R ( − , I ) {\displaystyle {\textrm {Hom}}_{R}(-,I)} is...

_{\parallel }} (see (13) and (14)). They are produced from the inner contravariant components F 1 , … , F n {\displaystyle F^{1},\,\ldots ,\,F^{n}} of...

abelian groups over X {\displaystyle X} , then the Verdier dual is a contravariant functor D : D b ( X ) → D b ( X ) {\displaystyle D\colon D^{b}(X)\to...

For a S ′ {\displaystyle S'} -scheme X {\displaystyle X} , if the contravariant functor Res S ′ / S ⁡ ( X ) : S c h / S o p → S e t , T ↦ Hom S ′ ⁡...

equation leads to the notion of a generalized cohomology theory; i.e., a contravariant functor from the category of spaces to the category of abelian groups...

multicategories, that simplicial sets have to categories. A dendroidal set is a contravariant functor from Ω to sets, where Ω is the tree category consisting of finite...

inclusions as morphisms, ⁠ O X {\displaystyle {\mathcal {O}}_{X}} ⁠ is a contravariant functor from ⁠ U {\displaystyle {\mathcal {U}}} ⁠ to the category of...

a continuous function from S(B) to S(A). In other words, there is a contravariant functor that gives an equivalence between the categories. The inverse...

\,\phi ^{*}(f)=f\circ \phi ,} then, Γ {\displaystyle \Gamma } is a contravariant functor from the category of affine algebraic sets with regular maps...

This is then a left R-module, and the *-operation yields a faithful contravariant functor from right R-modules to left R-modules. Every module of the...

component of the velocity field and V α {\displaystyle V^{\alpha }} be the contravariant components of the tangential velocity projection. Let ∇ α {\displaystyle...

light in free space. These relativistic equations can be written in contravariant form as ◻ A μ = 0 {\displaystyle \Box A^{\mu }=0} where the electromagnetic...

operators, and the components X μ {\displaystyle X^{\mu }} are often called contravariant components. This follows the standard notational conventions for sections...

of the general linear group GL(V) on the space of symmetric rank 2 contravariant tensors S2V∗ and classifies each orbit. The number v (resp. p) is the...

{\displaystyle A^{i_{q}}{}_{k_{q}}} is the transition matrix for the contravariant indices, B l p j p {\displaystyle B^{l_{p}}{}_{j_{p}}} is the transition...

at that point may be constructed. Vectors (sometimes referred to as contravariant vectors) are defined as elements of the tangent space and covectors...

{\displaystyle \mathbf {g} ^{1},\mathbf {g} ^{2},\mathbf {g} ^{3}} are the contravariant basis vectors in a curvilinear coordinate system, with coordinates of...

reverse operation, and I is the volume element (pseudoscalar) in the contravariant basis ( d x μ {\displaystyle dx^{\mu }} relating to e → μ {\displaystyle...

The first derivatives x ˙ i {\displaystyle {\dot {x}}^{i}} and the contravariant components of the local 3-velocity v i {\displaystyle v^{i}} are related...

superscripts of the basis functionals are not exponents but are instead contravariant indices. A linear functional u ~ {\displaystyle {\tilde {u}}} belonging...

using the non-generic System.Array type. However, since arrays are contravariant, the casting would not be type safe, and the compiler would be unable...

\mathrm {Set} ^{C^{op}}} (i.e. the functor category consisting of all contravariant functors from C {\displaystyle C} to S e t {\displaystyle \mathrm {Set}...

completely analogous way, one can also consider the category of all contravariant functors from C {\displaystyle C} to D {\displaystyle D} ; we write...

writing T k ( V ) {\displaystyle {\mathcal {T}}^{k}(V)} to denote the contravariant k-tensors on V {\displaystyle V} and T k ( V ) {\displaystyle {\mathcal...

direct check on the Kerr metric involves cumbersome calculations, the contravariant components ⁠ g i k {\displaystyle g^{ik}} ⁠ of the metric tensor in...

and cohomology is that in cohomology the chain complexes depend in a contravariant manner on X, and that therefore the homology groups (which are called...

g^{\mu }={\bar {\mathsf {h}}}(e^{\mu })} for a covariant vector and contravariant vector in a curved spacetime, where now the unit vectors { e μ } {\displaystyle...

category theory, Yoneda's lemma says that, given a category C, the contravariant functor C → P ( C ) = Fct ⁡ ( C op , S e t s ) , X ↦ Mor ⁡ ( − , X )...

of various types of stress tensor (i.e., the associated covariant, contravariant and mixed components of Cauchy stress) and their linear combinations...

bundles then corresponds to the inverse image of sheaves, which is a contravariant functor. A sheaf, however, is more naturally a covariant object, since...

the identity morphism. More concretely, let h {\displaystyle h} be a contravariant functor h : ( S c h / S ) o p → S e t s {\displaystyle h:(Sch/S)^{op}\to...

^{2}\setminus \{0\}\to \mathbb {R} ^{2}.} Borel−Moore homology is a contravariant functor with respect to inclusions of open subsets. That is, for U open...

(respectively right) rigid, and X* = X−1. The operation of taking duals gives a contravariant functor on a rigid category. One important application of rigidity is...

done above ( q i {\displaystyle q_{i}} ). The superscript conveys the contravariant transformation properties of the generalized coordinates, and does not...

algebraic closure of F p {\displaystyle \mathbb {F} _{p}} induces a contravariant equivalence from the homotopy category of connected p {\displaystyle...

to use either Ω or J. However, they have a different meaning: J is contravariant and Ω is covariant. The matrix Ω corresponds to the Lagrange brackets...

{e} ).} Much like the curvature, it can be shown that Θ behaves as a contravariant tensor under a change in frame: Θ i ( e g ) = ∑ j g j i Θ j ( e ) ....

Imperative Supported Function type intersection Distributive, co- and contravariant function type subtyping Java Yes Imperative Object-oriented Supported...

{\displaystyle hf=g} , i.e. such that the following diagram commutes: The contravariant Hom functor Hom ⁡ ( − , Q ) {\displaystyle \operatorname {Hom} (-,Q)}...

(\cdot )^{+}} and ( ⋅ ) + {\displaystyle (\cdot )_{+}} become a pair of contravariant functors between the category of general frames, and the category of...

f_{x}:(E_{1})_{x}\to (E_{2})_{x}.} Accordingly, the dual bundle operation defines a contravariant functor from the category of vector bundles and their morphisms to itself...

dual of vector spaces and the transpose of linear maps is therefore a contravariant functor from the category of vector spaces over K {\displaystyle {\mathcal...

and D, assigning each object and morphism to itself. If F and G are contravariant functors one speaks of a duality of categories instead. One often does...

respectively. If F {\displaystyle F} is not a (covariant) functor but a contravariant functor then by definition it induces morphisms in the opposite direction:...

upon changing the coordinate system, transform as the components of a contravariant vector. This discovery was the real beginning of tensor analysis. In...

^{j}.} In other terms, Θ is a tensor of type (1, 2) (carrying one contravariant and two covariant indices). Alternatively, the solder form can be characterized...

wrong way": it is a covariant map between objects associated with a contravariant functor. Gysin proved that the following is a long exact sequence ⋯...

Conversions from method groups to delegate types are covariant and contravariant in return and parameter types, respectively. Example: string status...

remaining n − k have the opposite sign. The correspondence between contravariant and covariant tensors makes a tensor calculus on pseudo-Riemannian manifolds...

function field in one variable over R. M is a contravariant functor and yields a duality (contravariant equivalence) between the category of compact connected...

the dimension of the manifold. Additionally, we can take the trace of contravariant tensors with respect to g {\displaystyle g} as follows (and similarly...

to the category of affine schemes. The equivalence is given by the contravariant functor Spec which sends a commutative ring to its spectrum, an affine...

this is a question of whether a certain functor is representable: the contravariant functor from the homotopy category to the category of sets, defined...

{\displaystyle \Rightarrow :\mathbf {Coh} ^{2}\to \mathbf {Coh} } , contravariant in the first argument and covariant in the second argument, such that...

rank on M with the monoid operation given by direct sum. This gives a contravariant functor from manifolds to abelian groups. This functor is studied and...

preserves all three classes of the Kan–Quillen model structure Co- and contravariant model structure, which can be induced by the Kan–Quillen model structure...

covariant/contravariant behavior is ignored entirely. Instead, contravariant components is implied by context. Hence for an n times contravariant tensor:...

f(xC). This definition had several difficulties: The construction acts contravariantly producing a dual of a continuous map rather than a generalization....

relativity, even with just a glance at the equations—using covariant and contravariant four-vectors and tensors. This can be done using the EM tensor F, or...

Quillen adjunctions between the injective model structures. Co- and contravariant model structure, induced model structures on slice categories Lurie...

contragredient or smooth dual of ( ρ , V ) {\displaystyle (\rho ,V)} . The contravariant functor ( ρ , V ) ↦ ( ρ ~ , V ~ ) {\displaystyle (\rho ,V)\mapsto ({\widetilde...

and E have topologies, to only consider continuous representations. A contravariant formalism can also be defined. In this case, the functor used is D B...

Then D ( f ) ↦ M [ f − 1 ] {\displaystyle D(f)\mapsto M[f^{-1}]} is a contravariant functor from the category whose objects are the sets D(f) and morphisms...

open subspaces of M, and the morphisms are inclusion maps. Let F be a contravariant functor from the category O(M) to the category Top of topological spaces...

derivative ∂ ; μ e ν b {\displaystyle \partial _{;\mu }e^{\nu b}} of the contravariant vector e ν b {\displaystyle e^{\nu b}} . The spin connection may be...

category ( A ∖ R ) / B {\displaystyle (A\backslash R)/B} since it gives a contravariant functor with values in Abelian groups. André–Quillen cohomology Cotangent...

{\displaystyle (\phi _{1}\circ \phi _{2})^{*}=\phi _{2}^{*}\phi _{1}^{*}} ( contravariant ) ϕ ∗ f = f ∘ ϕ {\displaystyle \phi ^{*}f=f\circ \phi } for f ∈ Ω 0...

several researchers introduced, among other things, covariantly and contravariantly finite subcategories of the category of finitely generated modules...

− γ α β {\displaystyle -\gamma _{\alpha \beta }} is inverse to the contravariant 3-dimensional tensor g α β {\displaystyle g^{\alpha \beta }} . Indeed...

differentiation with respect to 4-position X μ {\displaystyle X^{\mu }} . The contravariant components are: ∂ = ∂ α = η α β ∂ β = ( ∂ 0 , ∂ 1 , ∂ 2 , ∂ 3 ) = (...

fixed commutative ring R {\displaystyle R} with a unit, there exists a contravariant functor from the fundamental groupoid of B {\displaystyle B} to the...

Vector elements Contravariant transformation law Covariant transformation law Coordinates x ¯ j = x i ( L ) i j = x i L i j {\displaystyle {\bar...

time-averaged four-velocity, ∂ x μ {\displaystyle \partial ^{x^{\mu }}} are the contravariant components of the four-gradient and   ∗ {\displaystyle \ ^{*}} denotes...

substitutions from Δ {\displaystyle \Delta } to Γ {\displaystyle \Gamma } . A contravariant functor from this category to Fam {\displaystyle \operatorname {Fam}...

in the category of abelian groups. (To be precise, the latter is a contravariant functor, which can be considered as a functor on the opposite category...