Cartan calculus of cubical forms in tangent categories

Abstract

We construct the Cartan calculus of differential cubical forms on an object $X$ of a cartesian tangent category with a scalar multiplication by a commutative ring object $R$. Borrowing the terminology of cubical singular homology, we define an (infinitesimal) cubical $n$-form to be a morphism $\omega: T^n X \to R$ on the iterated tangent bundle that is antisymmetric and $R$-linear in every factor of $T^n$. We call $\omega$ differential if, in addition, it is its own derivative in the fiber directions. (This notion of forms is a special case of the singular forms of Cruttwell and Lucyshyn-Wright.) We prove that the differential cubical forms are naturally equipped with the structure of a commutative differential graded algebra (CDGA). Then we show that, if the ring object has no $2$-torsion, this CDGA together with the Lie algebra of vector fields and the inner derivatives constitutes a Cartan calculus. It lies between their initial Cartan calculus of algebraic K\"ahler forms and the terminal one of Lie-Rinehart forms. We give a number of examples. In particular, on affine schemes over a field of characteristic other than two and on elastic diffeological spaces we retrieve the usual de Rham complex, which is generally different from the complex of Lie-Rinehart forms.