Logic and Concepts in the 2-category of Topoi
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866910959399862272 |
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| author | Di Liberti, Ivan Ye, Lingyuan |
| author_facet | Di Liberti, Ivan Ye, Lingyuan |
| contents | We use Kan injectivity to axiomatise concepts in the 2-category of topoi. We showcase the expressivity of this language through many examples, and we establish some aspects of the formal theory of Kan extension in this 2-category (pointwise Kan extensions, fully faithful morphisms, etc.). We use this technology to introduce fragments of geometric logic, and we accommodate essentially algebraic, disjunctive, regular, and coherent logic in our framework, together with some more exotic examples. We show that each fragment $\mathcal{H}$ in our sense identifies a lax-idempotent (relative) pseudomonad $\mathsf{T}^{\mathcal{H}}$ on $\mathsf{lex}$, the $2$-category of finitely complete categories. We show that the algebras for $\mathsf{T}^{\mathcal{H}}$ admit a notion of classifying topos, for which we deliver several Diaconescu-type results. The construction of classifying topoi allows us to define conceptually complete fragments of geometric logic. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_16690 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Logic and Concepts in the 2-category of Topoi Di Liberti, Ivan Ye, Lingyuan Logic Logic in Computer Science Category Theory 03B10, 03G30, 18B25, 18C10, 18F10, 18N10, 18D65, 18A15 We use Kan injectivity to axiomatise concepts in the 2-category of topoi. We showcase the expressivity of this language through many examples, and we establish some aspects of the formal theory of Kan extension in this 2-category (pointwise Kan extensions, fully faithful morphisms, etc.). We use this technology to introduce fragments of geometric logic, and we accommodate essentially algebraic, disjunctive, regular, and coherent logic in our framework, together with some more exotic examples. We show that each fragment $\mathcal{H}$ in our sense identifies a lax-idempotent (relative) pseudomonad $\mathsf{T}^{\mathcal{H}}$ on $\mathsf{lex}$, the $2$-category of finitely complete categories. We show that the algebras for $\mathsf{T}^{\mathcal{H}}$ admit a notion of classifying topos, for which we deliver several Diaconescu-type results. The construction of classifying topoi allows us to define conceptually complete fragments of geometric logic. |
| title | Logic and Concepts in the 2-category of Topoi |
| topic | Logic Logic in Computer Science Category Theory 03B10, 03G30, 18B25, 18C10, 18F10, 18N10, 18D65, 18A15 |
| url | https://arxiv.org/abs/2504.16690 |