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3 | 3 | main: |
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| 5 | + - title: Invertible cells in \(\omega\)-categories |
| 6 | + authors: Thibaut Benjamin, Ioannis Markakis |
| 7 | + journal: Preprint |
| 8 | + date: 2024 |
| 9 | + pdf: https://arxiv.org/pdf/2406.12345.pdf |
| 10 | + arxiv : https://arxiv.org/abs/2406.12345 |
| 11 | + abstract: We study coinductive invertibility of cells in weak \(\omega\)\-categories. We use the inductive presentation of weak \(\omega\)\-categories via an adjunction with the category of computads, and show that invertible cells are closed under all operations of $\omega$\-categories. Moreover, we give a simple criterion for invertibility in computads, together with an algorithm computing the data witnessing the invertibility, including the inverse, and the cancellation data. |
| 12 | + doi: https://doi.org/10.48550/arXiv.2406.12345 |
| 13 | + |
5 | 14 | - title: CaTT contexts are finite computads |
6 | 15 | authors: Thibaut Benjamin, Ioannis Markakis, Chiara Sarti |
7 | | - journal: 40th Mathematical Foundations of Programming Semantics |
| 16 | + journal: Proceedings of the 40th Conference in Mathematical Foundations of Programming Semantics |
8 | 17 | date: 2024 |
9 | 18 | pdf: https://arxiv.org/pdf/2405.00398.pdf |
10 | 19 | arxiv : https://arxiv.org/abs/2405.00398 |
11 | 20 | abstract: Two novel descriptions of weak \(\omega\)-categories have been recently proposed, using type-theoretic ideas. The first one is the dependent type theory CaTT whose models are \(\omega\)-categories. The second is a recursive description of a category of computads together with an adjunction to globular sets, such that the algebras for the induced monad are again \(\omega\)-categories. We compare the two descriptions by showing that there exits a fully faithful morphism of categories with families from the syntactic category of CaTT to the opposite of the category of computads, which gives an equivalence on the subcategory of finite computads. We derive a more direct connection between the category of models of CaTT and the category of algebras for the monad on globular sets, induced by the adjunction with computads. |
12 | 21 | doi: https://doi.org/10.48550/arXiv.2405.00398 |
13 | 22 |
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14 | 23 |
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15 | | - - title: Duality for weak \(\omega\)-categories |
| 24 | + - title: Opposites for weak \(\omega\)-categories and the suspension and hom adjunction |
16 | 25 | authors: Thibaut Benjamin, Ioannis Markakis |
17 | 26 | journal: Preprint |
18 | 27 | date: 2024 |
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37 | 46 | pdf: https://arxiv.org/pdf/2208.08719.pdf |
38 | 47 | arxiv : https://arxiv.org/abs/2208.08719 |
39 | 48 | abstract: We give a new description of computads for weak globular ω-categories by giving an explicit inductive definition of the free words. This yields a new understanding of computads, and allows a new definition of \(\omega\)-category that avoids the technology of globular operads. Our framework permits direct proofs of important results via structural induction, and we use this to give new proofs that every ω-category is equivalent to a free one, and that the category of computads with variable-to-variable maps is a presheaf topos, giving a direct description of the index category. We prove that our resulting definition of \(\omega\)-category agrees with that of Batanin and Leinster and that the induced notion of cofibrant replacement for \(\omega\)-categories coincides with that of Garner. |
40 | | - doi: https://doi.org/10.48550/arXiv.2208.08719 |
| 49 | + doi: https://doi.org/10.1016/j.aim.2024.109739 |
41 | 50 | code: https://github.com/jmarkakis/computads.agda |
42 | 51 |
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43 | 52 | - title: Computing minimal generating systems for some special toric ideals |
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