Geometric differentiation of simplicial manifolds
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , |
|---|---|
| Format: | Preprint |
| Publié: |
2026
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866915805456760832 |
|---|---|
| author | Cabrera, Alejandro del Hoyo, Matias |
| author_facet | Cabrera, Alejandro del Hoyo, Matias |
| contents | We provide a complete geometric solution to the problem of differentiating simplicial manifolds, extending classical Lie theory and complementing existing homotopical and formal approaches within a unifying framework. First, we establish a normal form theorem setting a system of compatible tubular neighborhoods. Building on this description, we identify a differentiating ideal in the algebra of cochains, prove that the quotient is semi-free, and interpret it as the Chevalley-Eilenberg algebra of the thus defined higher Lie algebroid. As an application, we introduce a higher version of the van Est map and prove a van Est isomorphism theorem in cohomology, under natural connectivity assumptions. Finally, we identify the algebraic mechanism underlying geometric differentiation as a monoidal refinement of the dual Dold-Kan correspondence, providing a conceptual explanation of the construction and relating it to earlier homotopical and functor-of-points approaches. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_09885 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Geometric differentiation of simplicial manifolds Cabrera, Alejandro del Hoyo, Matias Differential Geometry Algebraic Topology Category Theory Primary: 22A22. Secondary: 58H05, 17B55, 18G30 We provide a complete geometric solution to the problem of differentiating simplicial manifolds, extending classical Lie theory and complementing existing homotopical and formal approaches within a unifying framework. First, we establish a normal form theorem setting a system of compatible tubular neighborhoods. Building on this description, we identify a differentiating ideal in the algebra of cochains, prove that the quotient is semi-free, and interpret it as the Chevalley-Eilenberg algebra of the thus defined higher Lie algebroid. As an application, we introduce a higher version of the van Est map and prove a van Est isomorphism theorem in cohomology, under natural connectivity assumptions. Finally, we identify the algebraic mechanism underlying geometric differentiation as a monoidal refinement of the dual Dold-Kan correspondence, providing a conceptual explanation of the construction and relating it to earlier homotopical and functor-of-points approaches. |
| title | Geometric differentiation of simplicial manifolds |
| topic | Differential Geometry Algebraic Topology Category Theory Primary: 22A22. Secondary: 58H05, 17B55, 18G30 |
| url | https://arxiv.org/abs/2602.09885 |