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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. We prove that the differential cubical forms are naturally equipped with the structure of a commutative differential graded algebra. Then we show that, if R has no $2$-torsion, this CDGA together with the Lie algebra of vector fields and the inner derivatives constitutes a Cartan calculus, which lies between the initial Cartan calculus of algebraic Kaehler forms and the terminal one of Lie-Rinehart forms. We give a number of examples, including affine schemes and elastic diffeological spaces, and show that we recover the established notions of differential forms in all cases.

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Cite

BibTeXRIS
BibTeX
@misc{blohmann2026cartan,
  title = {{Cartan calculus of cubical forms in tangent categories}},
  author = {Christian Blohmann},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.05963},
  eprint = {2609.05963},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.05963},
}
RIS
TY  - UNPB
TI  - Cartan calculus of cubical forms in tangent categories
AU  - Christian Blohmann
PY  - 2026
DA  - 2026/09/25
PB  - arXiv
DO  - 10.48550/arxiv.2609.05963
UR  - https://arxiv.org/abs/2609.05963
AB  - 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. We prove that the differential cubical forms are naturally equipped with the structure of a commutative differential graded algebra. Then we show that, if R has no $2$-torsion, this CDGA together with the Lie algebra of vector fields and the inner derivatives constitutes a Cartan calculus, which lies between the initial Cartan calculus of algebraic Kaehler forms and the terminal one of Lie-Rinehart forms. We give a number of examples, including affine schemes and elastic diffeological spaces, and show that we recover the established notions of differential forms in all cases.
ER  -