Displayed Type Theory and Semi-Simplicial Types
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866918286619312128 |
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| author | Kolomatskaia, Astra Shulman, Michael |
| author_facet | Kolomatskaia, Astra Shulman, Michael |
| contents | We introduce Displayed Type Theory (dTT), a multi-modal homotopy type theory with discrete and simplicial modes. In the intended semantics, the discrete mode is interpreted by a model for an arbitrary $\infty$-topos, while the simplicial mode is interpreted by Reedy fibrant augmented semi-simplicial diagrams in that model. This simplicial structure is represented inside the theory by a primitive notion of display or dependency, guarded by modalities, yielding a partially-internal form of unary parametricity.
Using the display primitive, we then give a coinductive definition, at the simplicial mode, of a type $\mathsf{SST}$ of semi-simplicial types. Roughly speaking, a semi-simplicial type $X$ consists of a type $X_0$ together with, for each $x:X_0$, a displayed semi-simplicial type over $X$. This mimics how simplices can be generated geometrically through repeated cones, and is made possible by the display primitive at the simplicial mode. The discrete part of $\mathsf{SST}$ then yields the usual infinite indexed definition of semi-simplicial types, both semantically and syntactically. Thus, dTT enables working with semi-simplicial types in full semantic generality. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_18781 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Displayed Type Theory and Semi-Simplicial Types Kolomatskaia, Astra Shulman, Michael Category Theory Logic We introduce Displayed Type Theory (dTT), a multi-modal homotopy type theory with discrete and simplicial modes. In the intended semantics, the discrete mode is interpreted by a model for an arbitrary $\infty$-topos, while the simplicial mode is interpreted by Reedy fibrant augmented semi-simplicial diagrams in that model. This simplicial structure is represented inside the theory by a primitive notion of display or dependency, guarded by modalities, yielding a partially-internal form of unary parametricity. Using the display primitive, we then give a coinductive definition, at the simplicial mode, of a type $\mathsf{SST}$ of semi-simplicial types. Roughly speaking, a semi-simplicial type $X$ consists of a type $X_0$ together with, for each $x:X_0$, a displayed semi-simplicial type over $X$. This mimics how simplices can be generated geometrically through repeated cones, and is made possible by the display primitive at the simplicial mode. The discrete part of $\mathsf{SST}$ then yields the usual infinite indexed definition of semi-simplicial types, both semantically and syntactically. Thus, dTT enables working with semi-simplicial types in full semantic generality. |
| title | Displayed Type Theory and Semi-Simplicial Types |
| topic | Category Theory Logic |
| url | https://arxiv.org/abs/2311.18781 |