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.