Should I delete this question and repost it on MSE?

Dmitri thank you for the reference. It looks like it's got other stuff I've been wanting to learn about as well.

Also, thank you both, that has answered my question :)

By baby Rudin, you mean "Principles..." right? I don't hate coordinates!!! Coordinates help us to visualize and understand things, and coordinate-free is another way. I don't think of one as better than the other, just different points of view, different organizations for calculations. Just as some coordinate systems facilitate calculations over other coordinate (e.g. spherical vs rectilinear)

Example: Given a manifold $M$, a vector bundle $E$ (or sheaf), and an open cover $(U_\alpha )$, one has the $\check{C}$ech complex associated to the cover. Each simplex has a bundle given by the restriction of the original bundle $E$ to the intersection corresponding to that simplex. I might have to $\check{C}$ech the details a little more, but I think this is an example.