We give a sheaf-theoretic definition of a connection $ \nabla$ on a vector bundle $ \pi \colon E \to M$, and then show how $ \nabla$ can be extended to a graded derivation

$$ d^{ \nabla} \colon \Omega^r_{M,E} \to \Omega^{r+1}_{M,E}.$$

This sheaf morphism behaves similarly to the standard exterior differential $d$, save for one key difference: it does not necessarily square to zero! In fact,

$$ d^{ \nabla} \circ d^{ \nabla} \xi= R^{\nabla} \wedge \xi,$$

and hence $d^{ \nabla} \circ d^{ \nabla} = 0$ if and only if $ \nabla$ is flat.

Comments and questions?