What is the higher dimensional infinitesimal groupoid of a manifold?

TL;DR

This paper extends Kapranov's construction of infinitesimal paths on a manifold to higher dimensions, revealing a full infinitesimal groupoid whose non-linear cohomology corresponds to the algebra of polyvector fields.

Contribution

It introduces a higher dimensional infinitesimal groupoid that generalizes previous models and connects to polyvector fields' algebra.

Findings

The full infinitesimal groupoid encodes higher dimensional infinitesimal objects.

The non-linear cohomology of this groupoid is the algebra of polyvector fields.

Extension of Kapranov's original construction to include contractions of infinitesimal loops.

Abstract

The construction (by Kapranov) of the space of infinitesimal paths on a manifold is extended to include higher dimensional infinitesimal objects, encoding contractions of infinitesimal loops. This full infinitesimal groupoid is shown to have the algebra of polyvector fields as its non-linear cohomology.