Weighted algebraic topology, II (Real valued metrics)
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arXiv
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866914562391932928 |
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| author | Grandis, Marco |
| author_facet | Grandis, Marco |
| contents | Extending the `metric spaces' of Lawvere, we study `real metrics', with values in the extended real line. Formally, this ordered set is a symmetric monoidal closed category, and our structures are enriched categories on the latter. Concretely, the present goal is measuring `profits' and `losses' of a process, in any sense - possibly related to energy, or a variable in any science. In particular, linear real metrics derive from a potential function.
This article is Part II in a series devoted to `weighted algebraic topology' - an enriched version of directed algebraic topology, where paths are measured. Part III will introduce a finer framework, more adequate to `quotient spaces' (as the spheres) and better related to topology. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_03487 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Weighted algebraic topology, II (Real valued metrics) Grandis, Marco Algebraic Topology Category Theory 54E35, 55Uxx, 18D20 Extending the `metric spaces' of Lawvere, we study `real metrics', with values in the extended real line. Formally, this ordered set is a symmetric monoidal closed category, and our structures are enriched categories on the latter. Concretely, the present goal is measuring `profits' and `losses' of a process, in any sense - possibly related to energy, or a variable in any science. In particular, linear real metrics derive from a potential function. This article is Part II in a series devoted to `weighted algebraic topology' - an enriched version of directed algebraic topology, where paths are measured. Part III will introduce a finer framework, more adequate to `quotient spaces' (as the spheres) and better related to topology. |
| title | Weighted algebraic topology, II (Real valued metrics) |
| topic | Algebraic Topology Category Theory 54E35, 55Uxx, 18D20 |
| url | https://arxiv.org/abs/2605.03487 |