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

[mnb96]

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]

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]

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.

 

[mnb96]

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]

I have never heard of invariance as a category theoretic notion in the same sense as covariance and contravariance. In fact just googling for invariance and tensors leads me only to the notions of invariants or of something being invariant under co-ordinate change. But these are not directly related to covariance etc.: at best one can say that invariants are things that are preserved by equivalences of categories (which are given by functors - the co/contra thing as I say is a red herring).