  A Q-algebroid is a Lie superalgebroid equipped with a compatible homological
vector field and is the infinitesimal object corresponding to a Q-groupoid. We
associate to every Q-algebroid a double complex. As a special case, we define
the BRST model of a Lie algebroid, which generalizes the BRST model for
equivariant cohomology. We extend to this setting the Mathai-Quillen-Kalkman
isomorphism of the BRST and Weil models, and we suggest a definition of a basic
subcomplex which, however, requires a choice of a connection. Other examples
include Roytenberg's homological double of a Lie bialgebroid, Ginzburg's model
of equivariant Lie algebroid cohomology, the double of a Lie algebroid matched
pair, and Q-algebroids arising from lifted actions on Courant algebroids.
