Combinatorial Differential Forms

Abstract

We extend to a scheme-theoretic context the notion of a combinatorial differential form, due to A.Kock in the framework of synthetic differential geometry. We show that group-valued combinatorial forms on a scheme may be identified, under very general hypotheses, with traditional Lie algebra-valued differential forms, and that their Lie algebra structure can be recovered from first principles. Some basic results from differential geometry (Maurer-Cartan equation, Bianchi identity), as well as a higher analogue, are obtained by exploiting this identification.