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| Format: | Preprint |
| Published: |
2025
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| Online Access: | https://arxiv.org/abs/2501.10672 |
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| _version_ | 1866914391650205696 |
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| author | Ortiz-Muñoz, Andrés |
| author_facet | Ortiz-Muñoz, Andrés |
| contents | We explore connections between homotopy type theory and information theory through homotopy cardinality. We define probability types and random variable types, prove that homotopy cardinality respects dependent sums under truncation and decidability hypotheses, and show that it does not respect dependent products in general. Using the power series expansion of the logarithm, expressed type-theoretically through deloopings of finite cyclic groups, we formulate Shannon entropy as the homotopy cardinality of a type and derive the chain rule for entropy under a trivial-action hypothesis. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_10672 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Homotopy Cardinality and Entropy Ortiz-Muñoz, Andrés Category Theory Information Theory Mathematical Physics 94A17, 03B38, 18B40 We explore connections between homotopy type theory and information theory through homotopy cardinality. We define probability types and random variable types, prove that homotopy cardinality respects dependent sums under truncation and decidability hypotheses, and show that it does not respect dependent products in general. Using the power series expansion of the logarithm, expressed type-theoretically through deloopings of finite cyclic groups, we formulate Shannon entropy as the homotopy cardinality of a type and derive the chain rule for entropy under a trivial-action hypothesis. |
| title | Homotopy Cardinality and Entropy |
| topic | Category Theory Information Theory Mathematical Physics 94A17, 03B38, 18B40 |
| url | https://arxiv.org/abs/2501.10672 |