  Working over a Lie groupoid, we establish a functorial correspondence between
representations up to homotopy and simplicial vector bundles equipped with a
cleavage. Our result, motivated by earlier work of Gracia-Saz and Mehta, can be
seen both as a relative version of the classical Dold-Kan correspondence, for
which we provide new geometric formulas, and also as a higher generalization of
the Grothendieck construction in a linear algebraic, smooth setting. We draw
connections to Vaintrob's theory of A-modules and to Lurie's homotopy colimit
equivalence, previously revisited by Heuts and Moerdijk. The ideas presented
here relate diverse viewpoints and have several applications, we comment on
some of them and outline our plans for future investigations.

Category Theory;Differential Geometry