Polynomial functors in π-clans for the semantics of type theory
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866917251242786816 |
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| author | Hua, Joseph Xu, Yiming |
| author_facet | Hua, Joseph Xu, Yiming |
| contents | The category of contexts underlying a model of Martin-Löf type theory with Unit-, $Σ$-, and $Π$-types need not be locally Cartesian closed, but is necessarily a $π$-clan. We exploit this $π$-clan structure to build the theory of polynomial functors. This paper presents two equivalent notions of strict semantics for MLTT in this weaker setting, respectively "elementary models" - reformulating categories with families - and "algebraic models" - reformulating natural models. These components fit into a practical sequence of steps for constructing models of MLTT: building an elementary model, extracting a $π$-clan from the elementary model, and then using polynomial functors built on the $π$-clan structure to convert the elementary model into an algebraic one. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_05689 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Polynomial functors in π-clans for the semantics of type theory Hua, Joseph Xu, Yiming Category Theory The category of contexts underlying a model of Martin-Löf type theory with Unit-, $Σ$-, and $Π$-types need not be locally Cartesian closed, but is necessarily a $π$-clan. We exploit this $π$-clan structure to build the theory of polynomial functors. This paper presents two equivalent notions of strict semantics for MLTT in this weaker setting, respectively "elementary models" - reformulating categories with families - and "algebraic models" - reformulating natural models. These components fit into a practical sequence of steps for constructing models of MLTT: building an elementary model, extracting a $π$-clan from the elementary model, and then using polynomial functors built on the $π$-clan structure to convert the elementary model into an algebraic one. |
| title | Polynomial functors in π-clans for the semantics of type theory |
| topic | Category Theory |
| url | https://arxiv.org/abs/2602.05689 |