I am currently reading up on Special Relativity, but most physics textbooks are very sketchy about how contravariance/covariance is defined and often just show how to convert between contravariant and covariant components of a vector via the metric tensor (all of which terms are used imprecisely). If someone has got a good reference that rigorously defines these concepts, but does not require a maths degree, and presents clear interpretations for the typical physics applications (SR and GR), please post it in a comment. This question, however, is more specific. I want to keep my understanding as rigorous as possible, so please correct all statements that are imprecise from a maths standpoint.
As I understand it, manifolds and vector spaces are two different things. Both are based on sets, but the additional structure imposed on those sets (via the corresponding axioms) is different. There may, however, be sets such as $\mathbb{R}^n$ that can be used with either structure, so can be considered in one way or another.
At each point $x$ in a differentiable manifold $M$, a tangent space can be constructed and the elements of this tangent space form a vector space. So a manifold by itself is not a vector space, but certain manifolds allow (different) vector spaces $V_x$ to be defined at each (different) point $x$.
My question is now: Does the concept of contravariance/covariance only apply to these special vector spaces that are defined in conjunction with a manifold $M$ or does it apply generally to any abstract vector space?
Manifolds are constructed locally and then patched together globally. Thus to understand to covariance and contravariance globally, we can understand it first locally, and in fact it suffices to understand it over a point, and this means we need only understand it in terms of vector spaces.
Given a vector space $V$, it turns out that there is a cofunctor:
Here, $Bs(V)$ is the category of all bases of $V$, with morphisms change of bases. So for example, $A:e \rightarrow e'$ is a morphism in this category, where $e,e$ are bases of $V$; and it turns out that $A$ can be represented (not in the technical sense), by $A \in GL(n)$.
Whereas, $CoFrm(V)$ is the category of coframes of $V$ (this is not standard terminology), where a coframe is a map $E:V \rightarrow K^n$, where $K$ is our ground field. (I take a frame, to be the opposite sense; that is a map from $K^n \rightarrow V$. Often these two senses are elided leading to much confusion about frames, bases and the like. Again the terminology is non-standard).
The functor exhibits the fact that coframes vary contravariantly wrt change of bases. And makes precise, the notion of contravariance beloved by introductory books on GR. although, Sean Carroll, the physicist thinks contrvariance as rather an out-dated notion; I think it has been a valuable input into the notions of variance in category theory. More physically speaking, the notion of a G-bundle, and more precisely, the notion of a principal G-bundle, where the group action acts on the bundle on the right, actually arises from this idea - which one can see by specialsing the bundle to be over a point.