A Unified View on the Functorial Nerve Theorem and its Variations
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2022
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| _version_ | 1866910980744675328 |
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| author | Bauer, Ulrich Kerber, Michael Roll, Fabian Rolle, Alexander |
| author_facet | Bauer, Ulrich Kerber, Michael Roll, Fabian Rolle, Alexander |
| contents | The nerve theorem is a basic result of algebraic topology that plays a central role in computational and applied aspects of the subject. In topological data analysis, one often needs a nerve theorem that is functorial in an appropriate sense, and furthermore one often needs a nerve theorem for closed covers as well as for open covers. While the techniques for proving such functorial nerve theorems have long been available, there is unfortunately no general-purpose, explicit treatment of this topic in the literature. We address this by proving a variety of functorial nerve theorems. First, we show how one can use elementary techniques to prove nerve theorems for covers by closed convex sets in Euclidean space, and for covers of a simplicial complex by subcomplexes. Then, we establish a more general, "unified" nerve theorem that subsumes many of the variants, using standard techniques from abstract homotopy theory. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2203_03571 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | A Unified View on the Functorial Nerve Theorem and its Variations Bauer, Ulrich Kerber, Michael Roll, Fabian Rolle, Alexander Algebraic Topology 55N31 (Primary) 55-02, 55P10 (Secondary) The nerve theorem is a basic result of algebraic topology that plays a central role in computational and applied aspects of the subject. In topological data analysis, one often needs a nerve theorem that is functorial in an appropriate sense, and furthermore one often needs a nerve theorem for closed covers as well as for open covers. While the techniques for proving such functorial nerve theorems have long been available, there is unfortunately no general-purpose, explicit treatment of this topic in the literature. We address this by proving a variety of functorial nerve theorems. First, we show how one can use elementary techniques to prove nerve theorems for covers by closed convex sets in Euclidean space, and for covers of a simplicial complex by subcomplexes. Then, we establish a more general, "unified" nerve theorem that subsumes many of the variants, using standard techniques from abstract homotopy theory. |
| title | A Unified View on the Functorial Nerve Theorem and its Variations |
| topic | Algebraic Topology 55N31 (Primary) 55-02, 55P10 (Secondary) |
| url | https://arxiv.org/abs/2203.03571 |