  We construct a graded Lie algebra $\mathcal{E}$ in which the Maurer-Cartan
equation is equivalent to the vacuum Einstein equations. The gauge groupoid is
the groupoid of rank 4 real vector bundles with a conformal inner product, over
a 4-dimensional base manifold, and the graded Lie algebra construction is a
functor out of this groupoid. As usual, each Maurer-Cartan element in
$\mathcal{E}^1$ yields a differential on $\mathcal{E}$. Its first homology is
linearized gravity about that element. We introduce a gauge-fixing algorithm
that generates, for each gauge object $G$, a contraction to a much smaller
complex whose modules are the kernels of linear, symmetric hyperbolic partial
differential operators. This contraction opens the way to the application of
homological algebra to the analysis of the vacuum Einstein equations. We view
general relativity, at least at the perturbative level, as an instance of
`homological PDE' at the crossroads of algebra and analysis.
