# Co/contra/in-variance of tensors in abstract algebra

Hello,
The concept of contravariance, covariance and invariance are commonly used in the domain of Tensor Calculus. However I have heard that such concepts are more abstractly defined (perhaps) in cathegory theory.

Could someone explain shortly the connection between the abstract definitions of contravariance/covariance and the definitions used in tensor calculus?

matt grime
Homework Helper
Co- and contravariance refer to what happens to the direction of arrows under a functor.

Recall that a category has objects and morphisms between them, commonly viewed as arrows. A functor is a map from one category to another. It sends objects to objects and morphisms to morphisms, or arrows to arrows. A covariant functor preserves the direction of arrows, a contravariant one reverses them.

The obvious candidate is the category of (finite dimensional) vector spaces. The functor sending V to V*, the dual space, is contravariant as a map

f:V-->W

has a dual map

f*:W*-->V*

This is now seen as largely unnecessary, since a contravariant functor from A to B is just a covariant functor from A to B^{op}.

If you want another canonical example, take Hom(X,-) and Hom(-,X) as your functors. One is contravariant, the other covariant.

Hurkyl
Staff Emeritus
Gold Member
The relevance to tensor calculus over a manifold is that the typical objects of study are built from the tangent bundle and cotangent bundle. "Tangent bundle" is a covariant functor from manifolds (as is all of its tensor powers), and "cotangent bundle" is a contravariant functor (as is all of its tensor powers).

For example, the standard basis for the tangent bundle of R (along with the dual basis for its cotangent bundle) yields the following:

1. Every smooth curve f : R --> M has a canonical tangent vector at each of its points. (Multiple ones, of course, if f(a) = f(b) ever holds for $a \neq b$)

2. Every smooth function f : M --> R determines a canonical covector field on M.

I believe the opposite naming convention for tensors was once used, but the one described above is in favor now.

Thanks,
your explanations on contra/co-variance were very clear.
However, since I don't have a strong background in category theory, I am still unable to see how/if the abstract concept of invariance directly arises from covariance-contravariance.

matt grime