Canonical differential calculi via functorial geometrization
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arXiv
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| Formato: | Preprint |
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2025
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| _version_ | 1866911610031833088 |
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| author | Flood, Keegan J. Lobbia, Gabriele Tendas, Giacomo |
| author_facet | Flood, Keegan J. Lobbia, Gabriele Tendas, Giacomo |
| contents | Given a category $\mathcal{E}$, we establish sufficient conditions on a faithful isofibration $\mathcal{E}\rightarrow\operatorname{Mon}(\mathcal{V})$ valued in the category of monoids internal to a monoidal additive category $\mathcal{V}$ such that $\mathcal{E}$ admits a canonical functor to the category of first order differential calculi in $\mathcal{V}$. Generalizing the procedure of extending a first order differential calculus to its maximal prolongation to this setting, we obtain a canonical functor from $\mathcal{E}$ to the category of differential calculi in $\mathcal{V}$. This yields a simultaneous generalization of the de Rham complex on $C^{\infty}$-rings, the Kähler differentials on commutative algebras, and the universal differential calculus on associative algebras. As a consequence, such categories $\mathcal{E}$ admit natural analogues of the notions of smooth map and diffeomorphism, as well as a functorial de Rham theory. Moreover, whenever two such faithful isofibrations to $\operatorname{Mon}(\mathcal{V})$ factor suitably, their corresponding de Rham functors are related via a comparison map. Developing this theory requires first extending the noncommutative geometry formalism of differential calculi from associative algebras to the setting of monoids internal to monoidal additive categories. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_20742 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Canonical differential calculi via functorial geometrization Flood, Keegan J. Lobbia, Gabriele Tendas, Giacomo Category Theory Quantum Algebra Rings and Algebras Primary 18M05, 16D90, 18C40, 58B34, 58B32, 16E45, Secondary 18M60, 18C35, 18C40 Given a category $\mathcal{E}$, we establish sufficient conditions on a faithful isofibration $\mathcal{E}\rightarrow\operatorname{Mon}(\mathcal{V})$ valued in the category of monoids internal to a monoidal additive category $\mathcal{V}$ such that $\mathcal{E}$ admits a canonical functor to the category of first order differential calculi in $\mathcal{V}$. Generalizing the procedure of extending a first order differential calculus to its maximal prolongation to this setting, we obtain a canonical functor from $\mathcal{E}$ to the category of differential calculi in $\mathcal{V}$. This yields a simultaneous generalization of the de Rham complex on $C^{\infty}$-rings, the Kähler differentials on commutative algebras, and the universal differential calculus on associative algebras. As a consequence, such categories $\mathcal{E}$ admit natural analogues of the notions of smooth map and diffeomorphism, as well as a functorial de Rham theory. Moreover, whenever two such faithful isofibrations to $\operatorname{Mon}(\mathcal{V})$ factor suitably, their corresponding de Rham functors are related via a comparison map. Developing this theory requires first extending the noncommutative geometry formalism of differential calculi from associative algebras to the setting of monoids internal to monoidal additive categories. |
| title | Canonical differential calculi via functorial geometrization |
| topic | Category Theory Quantum Algebra Rings and Algebras Primary 18M05, 16D90, 18C40, 58B34, 58B32, 16E45, Secondary 18M60, 18C35, 18C40 |
| url | https://arxiv.org/abs/2512.20742 |