Thickening of the diagonal and interleaving distance
| hal.structure.identifier | Centre de droit comparé du travail et de la sécurité sociale [COMPTRASEC] | |
| dc.contributor.author | PETIT, Francois | |
| hal.structure.identifier | Sorbonne Université [SU] | |
| dc.contributor.author | SCHAPIRA, Pierre | |
| dc.description.abstractEn | Given a topological space $X$, a thickening kernel is a monoidal presheaf on $(\mathbb{R}_{\geq0},+)$ with values in the monoidal category of derived kernels on $X$. A bi-thickening kernel is defined on $(\mathbb{R},+)$. To such a thickening kernel, one naturally associates an interleaving distance on the derived category of sheaves on $X$. We prove that a thickening kernel exists and is unique as soon as it is defined on an interval containing $0$, allowing us to construct (bi-)thickenings in two different situations. First, when $X$ is a ``good'' metric space, starting with small usual thickenings of the diagonal. The associated interleaving distance satisfies the stability property and Lipschitz kernels give rise to Lipschitz maps. Second, by using [GKS12], when $X$ is a manifold and one is given a non-positive Hamiltonian isotopy on the cotangent bundle. In case $X$ is a complete Riemannian manifold having a strictly positive convexity radius, we prove that it is a good metric space and that the two bi-thickening kernels of the diagonal, one associated with the distance, the other with the geodesic flow, coincide. | |
| dc.language.iso | en | |
| dc.title.en | Thickening of the diagonal and interleaving distance | |
| dc.type | Document de travail - Pré-publication | |
| dc.subject.hal | Mathématiques [math] | |
| dc.identifier.arxiv | 2006.13150 | |
| hal.identifier | hal-03839986 | |
| hal.version | 1 | |
| hal.origin.link | https://hal.archives-ouvertes.fr//hal-03839986v1 | |
| bordeaux.COinS | ctx_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.au=PETIT,%20Francois&SCHAPIRA,%20Pierre&rft.genre=preprint |
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