Homologie polygraphique des systèmes locaux
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arXiv
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| Format: | Preprint |
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2023
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| author | Guetta, Léonard Maltsiniotis, Georges |
| author_facet | Guetta, Léonard Maltsiniotis, Georges |
| contents | In this article, we introduce a notion of polygraphic homology of a strict $ω$-category with coefficients in a local system, generalizing the polygraphic homology with coefficients in $\mathbb Z$, introduced by François Métayer. We show that the homology of a simplicial set with coefficients in a local system coincides with the polygraphic homology of its image by the left adjoint of the Street nerve with coefficients in the corresponding local system. We define in this framework a comparison morphism between the polygraphic homology of a strict $ω$-category and the homology of its Street nerve, and we show that this morphism is an isomorphism for (1-)categories. This is not true for an arbitrary $ω$-category. Nevertheless, we conjecture that for an analogous construction in the framework of weak $ω$-categories ``à la Grothendieck'' we would always obtain an isomorphism. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_10466 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Homologie polygraphique des systèmes locaux Guetta, Léonard Maltsiniotis, Georges Algebraic Topology Category Theory 18G10, 18G15, 18G35, 18G90, 18N30, 18N40, 18N50, 18N65, 55N10, 55N25, 55U10, 55U15 In this article, we introduce a notion of polygraphic homology of a strict $ω$-category with coefficients in a local system, generalizing the polygraphic homology with coefficients in $\mathbb Z$, introduced by François Métayer. We show that the homology of a simplicial set with coefficients in a local system coincides with the polygraphic homology of its image by the left adjoint of the Street nerve with coefficients in the corresponding local system. We define in this framework a comparison morphism between the polygraphic homology of a strict $ω$-category and the homology of its Street nerve, and we show that this morphism is an isomorphism for (1-)categories. This is not true for an arbitrary $ω$-category. Nevertheless, we conjecture that for an analogous construction in the framework of weak $ω$-categories ``à la Grothendieck'' we would always obtain an isomorphism. |
| title | Homologie polygraphique des systèmes locaux |
| topic | Algebraic Topology Category Theory 18G10, 18G15, 18G35, 18G90, 18N30, 18N40, 18N50, 18N65, 55N10, 55N25, 55U10, 55U15 |
| url | https://arxiv.org/abs/2309.10466 |