  We present a novel proof technique to construct the Gelfand-Fuks spectral
sequence for diagonal Chevalley-Eilenberg cohomology of vector fields on a
smooth manifold, performing a local-to-global analysis through a notion of
generalized good covers from the theory of factorization algebras and
cosheaves. This approach yields a unified way to deal with the problem of
comparing "sheaf-like" data over different Cartesian powers of the manifold,
and is easily generalized to the study of other cohomology theories associated
to geometric objects. Independently, we lay out a detailed and easily
accessible exposition on the continuous Chevalley-Eilenberg cohomology of
formal vector fields and of vector fields on Euclidean space, modernizing and
elaborating on well-established literature on the subject by Bott, Fuks, and
Gelfand.

K-Theory and Homology;Differential Geometry