[jasomill]

A 1-form is a section[1] of the cotangent bundle[2] of a manifold. In other words, a rank 1 covariant tensor field.

At any given point p on an n-dimensional manifold, a 1-form defines an n-dimensional cotangent vector (in the language of bundles[3], a point in the fiber over p).

So how do we define fractions of sections or vectors?

In the article, Baez defines fractions of 2-forms on the plane as the pointwise ratio of coefficients of a basis vector, which he can do because, as he points out, the space of 2-forms at a point on a 2-dimensional manifold is a 1-dimensional vector space (more generally, for k-forms on an n-dimensional manifold, this dimension is n choose k, so only 1 for 0-forms [functions] and n-forms).

[1] https://mathworld.wolfram.com/BundleSection.html

[2] https://mathworld.wolfram.com/CotangentBundle.html

[3] https://mathworld.wolfram.com/FiberBundle.html