Welcome to the next installation of my series on local systems. In this post I’ll be talking about connections. This post should require less sophistication than the last few — no schemes, no functors — I’ll almost be coming at the subject afresh. There will be another post later, explaining how you might get to connections if you started out thinking about the infinitesimal site.

To start out with, let’s talk about derivatives; ordinary, single variable calculus derivatives. We have a function of a variable . Then the derivative of is the function . There are two directions in which we might want to generalize this idea. The first is to work with functions on a manifold, on a space which has no inherent coordinate system. This is the subject of your standard Calculus on Manifolds course, and I am going to assume that my readers are at least vaguely familiar with it. The second is to work, not with functions, but with sections of vector bundles. That’s our subject in this post.

So, let’s think about a vector bundle on the line , and let be a section of . If we want to define , we need to subtract and , two vectors which live in different fibers. To think of it another way, we need to distinguish between , the point in the fiber over , and , the constant function which assigns the same value at every point. Suppose that, for any , we had a local section of with ; we think of as a constant function. Then we could define .

A local system gives us the constant functions . (Indeed, in definitions A.2 and B.6, we took a local system to **be** the constant functions, along with the data of certain maps between them.) Today, we will take the fundamental object to be the operation of derivation, and see how to build everything else from it.