The Frolicher-Nijenhuis Calculus in Synthetic Differential Geometry

TL;DR

This paper demonstrates that the graded Jacobi identity of the Frolicher-Nijenhuis bracket naturally follows from the general Jacobi identity within the framework of synthetic differential geometry, linking classical algebraic structures to synthetic theory.

Contribution

It establishes a conceptual connection between the Frolicher-Nijenhuis calculus and synthetic differential geometry by deriving the graded Jacobi identity from the general Jacobi identity.

Findings

The graded Jacobi identity of the Frolicher-Nijenhuis bracket is a consequence of the general Jacobi identity.

Synthetic differential geometry provides a natural setting for understanding classical algebraic identities.

The approach unifies classical and synthetic perspectives on differential calculus.

Abstract

Just as the Jacobi identity of vector fields is a natural consequence of the general Jacobi identity of microcubes in synthetic differential geometry, it is to be shown in this paper that the graded Jacobi identity of the Frolicher-Nijenhuis bracket is also a natural consequence of the general Jacobi identity.