The classical Dold–Kan correspondence for simplicial objects in an abelian category is one of the cornerstones of homological algebra. When the abelian category is that of vector spaces, it gives a full identification between simplicial vector spaces and chain complexes of vector spaces vanishing in negative degrees. The Grothendieck construction for fibered categories, on the other hand, is a cornerstone of category theory. It relates the fibered category point of view with the pseudo-functor point of view and lies at the heart of the theory of stacks. Our main result can be understood as a far-reaching simultaneous generalization of both ideas within the contexts of linear algebra and differential geometry. In our result, simplicial vector spaces and chain complexes of vector spaces are replaced respectively by vector fibrations over a given (higher) Lie groupoid G and by representations up to homotopy of G. (Joint work with Matias del Hoyo.)