Categorical characterisations of quasi-isometric embeddings
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866916479381798912 |
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| author | Tang, Robert |
| author_facet | Tang, Robert |
| contents | We characterise the (closeness classes of) quasi-isometric embeddings as the regular monomorphisms in the coarsely Lipschitz category, formalising the notion that they are isomorphisms onto their image. Furthermore, we prove that the coarsely Lipschitz category is coregular, and hence admits an (Epi, RegMono)--orthogonal factorisation system. Consequently, quasi-isometric embeddings are equivalently characterised as the effective, strong, or extremal monomorphisms. Finally, we prove that the coarsely Lipschitz category is not coexact in the sense of Barr. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_08501 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Categorical characterisations of quasi-isometric embeddings Tang, Robert Metric Geometry Category Theory Group Theory Geometric Topology 51F30, 20F65 We characterise the (closeness classes of) quasi-isometric embeddings as the regular monomorphisms in the coarsely Lipschitz category, formalising the notion that they are isomorphisms onto their image. Furthermore, we prove that the coarsely Lipschitz category is coregular, and hence admits an (Epi, RegMono)--orthogonal factorisation system. Consequently, quasi-isometric embeddings are equivalently characterised as the effective, strong, or extremal monomorphisms. Finally, we prove that the coarsely Lipschitz category is not coexact in the sense of Barr. |
| title | Categorical characterisations of quasi-isometric embeddings |
| topic | Metric Geometry Category Theory Group Theory Geometric Topology 51F30, 20F65 |
| url | https://arxiv.org/abs/2411.08501 |