---
title: "Cartan calculus of cubical forms in tangent categories"
authors:
  - "Christian Blohmann"
year: 2026
venue: "arXiv:math.CT"
arxiv: "2609.05963"
doi: null
source: "https://arxiv.org/abs/2609.05963"
topics:
  - "curiosity"
  - "maths-foundations"
seed_rank: 1783
seed_batch: "curiosity-2026-09-11"
reviewed: "2026-09-11"
pool: "maths-foundations"
relevance_score: 10
lineage: tangent-categories-ddg
cites:
  - title: "Cartan calculus of cubical forms in tangent categories"
    url: "https://arxiv.org/abs/2609.05963"
    year: 2026
    arxiv: "2609.05963"
    doi: null
see:
  - "1582-the-lie-group-lie-algebra-correspondence-in-tangent-categories"
  - "1028-discrete-exterior-calculus"
  - "1056-reverse-derivative-categories"
  - "1051-finite-element-exterior-calculus"
---

# Cartan calculus of cubical forms in tangent categories

## One-sentence takeaway

Cartan calculus (exterior derivative, Lie derivative, interior product) for differential cubical forms, developed internally in a cartesian tangent category with scalars.

## Why it's lovely

Why you might love this: tangent categories already package "derivative" as structure; here the full Cartan calculus of forms arrives in cubical (not just simplicial/Koszul) clothing. Direct bridge from synthetic differential geometry to Discrete Exterior Calculus / FEEC instincts — and to Anoptic's geometry stack — without leaving the categorical language of 1582.

## Key ideas

- Cubical n-form: morphism TⁿX → R, antisymmetric and R-linear in each tangent factor.
- Differential forms: additionally equal to their own fibrewise derivative.
- Special case of Cruttwell–Lucyshyn-Wright singular forms.
- Full Cartan calculus developed in a cartesian tangent category with ring object R.
- Sits beside tangent-category Lie theory (1582), DEC (1028), and reverse-derivative categories (1056).

## Caveats

Synthetic/categorical; extracting a concrete mesh DEC algorithm still needs a discrete model. Cubical (vs exterior-algebra) presentation may feel unfamiliar at first.

## Links

- arXiv abs: https://arxiv.org/abs/2609.05963
- PDF: https://arxiv.org/pdf/2609.05963.pdf