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arXiv · 2609.05963

Cartan calculus of cubical forms in tangent categories

Christian Blohmann
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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 $ω: T^n X \to R$ on the iterated tangent bundle that is antisymmetric and $R$-linear in every factor of $T^n$. We call $ω$ 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 Kähler 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.

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Christian Blohmann. 2026-09-05. Cartan calculus of cubical forms in tangent categories. https://arxiv.org/abs/2609.05963

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