Co- and contravariance of vectors vs co- and contravariant functors

Question

I am wondering whether there is any connection between

1) the physicists use of the words co- and contravariant in the context of vectors and their transfomations

2) The notion of co- and contravariant functors.

If so,

i) What exactly are the functors involved?

ii) Is there some relation to the (contravariant) dual-functor *?

Greetings, Tobi

PS: I have no prior knowledge of differential topology, manifolds or multilinear algebra.

Answer 1

Yes, there's a connection. Briefly: if you're measuring "position" in a particular coordinate system, then there's a related "velocity space" coordinate system consisting of all possible velocity vectors that can occur throughout the points in your coordinate system.

Every smooth transformation from one coordinate system to another affects velocity vectors in a systematic (specifically, linear) way. (To see this effect, just apply the chain rule to compare the velocity vector in the old and new coordinate systems.) This predictable effect on velocity vectors induces a change of coordinates in the associated velocity space.

Thus, to each coordinate system, there's a related velocity coordinate system. And to each transformation between coordinate systems, there's an induced change of this velocity coordinate system. Finally, as it turns out, this relationship respects composition: the chain rule of calculus tells us that if we change coordinates twice in a row and compute a velocity vector in terms of new coordinates, it's the same as if we computed the velocity vector in the original space, then sent it through the two induced velocity coordinate changes.

• The "velocity space" construct is a functor $T$.
• $T$ sends the category of coordinate systems and smooth, invertible transformations to itself.
• $T$ sends every coordinate system $M$ to its associated velocity space coordinate system $TM$.
• $T$ sends every coordinate transformation $f:M\rightarrow N$ to an induced change between velocity coordinate systems $Tf: TM\rightarrow TN$.
• And the chain rule of calculus says that this induced change respects composition: $T(f\circ g) = T(f) \circ T(g)$.

That is the functor involved for velocities. What about quantities that vary the other way, such as the gradient? Velocities— the set of tangent vectors at each point — form a linear space. As such, we can consider the dual space consisting of the dual of tangent vectors. If we compute how a change in coordinates affects the dual space, we find that it systematically induces a linear transformation in the reverse direction:

$$T^*(f):TN^* \rightarrow TM^*$$

so quantities that vary the other way (such as the gradient) constitute a contravariant functor. The dual appears both in the vector space sense (linear dual space) and in the category theory sense (as a contravariant functor).

Details follow.

---

1. In differential geometry, you can think of a change in coordinates as smooth reversible transformation.

   (Actually, this is just as in calculus, where a legal change of variables is a function that is invertible and differentiable.)

2. Coordinate changes affect velocity vectors in a linear way. One interesting fact about coordinate transforms (i.e. smooth reversible transformations) is that they affect velocity vectors in a reliable, structured way. In particular, they affect velocity vectors in a linear way: if there's a curve traveling through point $p$ and its velocity vector there is $\vec{v}$, then when you change coordinates, the curve in the new space will be traveling through point $f(p)$ and its velocity vector there will be some linear function $L_p(\vec{v})$.

   I write the function with a subscript $p$ because this linear function will generally be different for different points $p$. But different velocity vectors at $p$ will all be transformed by the same linear function $L_p$.

3. In Euclidean space, velocities can point in any direction and the fact that coordinate transforms linearly affect velocity vectors is sort of pedestrian. The linearity is more remarkable for curved spaces, such as the surface of a sphere, where the set of possible velocity vectors at a point—the tangent vectors—are more interestingly constrained.

   

   Even on these curved, constrained coordinate systems, a smooth reversible change of coordinates always induces linear maps $L_p$ between the space of tangent vectors anchored at each point.

4. Another insight: Velocity-space is a coordinate space, too. If every position in your coordinate space can be located with $n$ coordinates, then every tangent (velocity) vector can be located with 2n coordinates: the first $n$ coordinates locate a position in space, and the next $n$ coordinates describe the components of the velocity at that point according to some pre-established basis (messy details excluded).

   In differential geometry, this velocity space is called the tangent bundle. It is the combined collection of all tangent vector spaces, where we had one space per point. If $M$ is the original coordinate space, we can write $TM$ as its tangent bundle— the velocity space for $M$.

5. What we have discovered is that a change of coordinates systems $M\rightarrow N$ gives rise to a family of linear maps $L_p : TM_p \rightarrow TN_p$, one linear map per point $p$.

   (Here, the notation $TM_p$ means the space of tangent vectors at a single point $p$.)

   In fact, this family of linear maps constitutes a coordinate transformation, this time between the velocity space $TM$ and the velocity space $TN$.

   Every coordinate change for position $f : M\rightarrow N$ gives rise to a coordinate change of velocity-space $Tf: TM \rightarrow TN$.

   And, according to the chain rule of calculus, the construct $Tf$ even respects composition: if $f$ and $g$ are successive coordinate transforms, then $g\circ f$ is a coordinate transform and according to the chain rule of calculus, we can confirm that $T(g\circ f) = Tg \circ Tf$.

6. Thus, this systematic relationship $T$ between how a coordinate change affects position and how it affects velocity behaves as a functor:

   • If $M$ is a "position" coordinate system, then $TM$ is its associated velocity coordinate system.
   • The velocity coordinate system $TM$ is the collection of all the tangent vector spaces for all the points in $M$.
   • If $f:M\rightarrow N$ is a transformation between coordinate systems,
   • Then the resulting linear effect of $f$ on the tangent spaces gives rise to a map $Tf : TM \rightarrow TN$ which describes a change in coordinates between the two velocity-spaces.
   • By the chain rule of calculus, $T(g\circ f) = T(g)\circ T(f)$, which shows that $T$ respects composition and is therefore genuinely a functor.

7. About covectors. The dual construct to $T$ requires only a little extra work: we earlier considered the linear space of tangent vectors at each point. Every change in coordinates affects the tangent vectors in a linear way, inducing a function $T$.

   But because every linear space gives rise to a linear dual space, we could have also considered associating each point with the dual of the space of tangent vectors. This dual space— which we could say is made up of tangent covectors— also systematically changes when you change the coordinates of your space.

   But here, the induced change is a backwards map: you can calculuate (e.g. for a simple change in coordinates like $f(x,y) = \langle 3x-2y,\, -x\rangle$) that a change $f:M\rightarrow N$ induces a linear map from covectors $TN^* \rightarrow TM^*$. In the reverse direction!

   Following the same reasoning as above, we find that "co-velocity space" is also a functor. It sends each coordinate space $M$ to the co-velocity space $TM^*$, and it sends each transformation of position coordinates $f:M\rightarrow N$ to a reverse change in coordinates $TN^*\rightarrow TM^*$. Hence "co-velocity" space is a dual functor.

8. Awkwardly, the terminology for category theory and for coordinate transforms uses the co- prefix in different (basically opposite) ways. But here's a chart to avoid confusion:

   • Quantities like velocity and the spatial gradient behave oppositely under coordinate changes.

   • One standard of comparison is relative to the coordinate axes: if your transformation lengthens the scale of the coordinate axes, velocity vectors proportionately shrink and gradient vectors proportionately lengthen.

   • Of course, another standard of comparison is relative to velocity vectors themselves: If your transformation lengthens velocity vectors, then it shrinks gradient vectors.
   • The standard reference point for coordinate transforms is the first one, the coordinate axes: coordinate transforms refer to quantities as "covariant" and "contravariant" relative to the coordinate axes. Thus, the gradient "covaries" and velocities "contravary" with respect to the axes. The gradient is a "vector" and velocities are "covectors" with respect to the coordinate axes.
   • But for ease of use with category theory, a different reference point might be nice. Category theory has covariant and contravariant functors. We had a covariant functor for velocity space and a contravariant functor for co-velocity space (e.g. the gradient), so it might be nice if we adopted the reference point relative to which velocity covaries and the gradient contra-varies. Alas, this convention is not standard in geometry.