2-classifiers via dense generators and Hofmann-Streicher universe in stacks
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910608774922240 |
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| author | Mesiti, Luca |
| author_facet | Mesiti, Luca |
| contents | We expand the theory of 2-classifiers, that are a 2-categorical generalization of subobject classifiers introduced by Weber. The idea is to upgrade monomorphisms to discrete opfibrations. We prove that the conditions of 2-classifier can be checked just on a dense generator. The study of what is classified by a 2-classifier is similarly reduced to a study over the objects that form a dense generator. We then apply our results to the cases of prestacks and stacks, where we can thus look just at the representables. We produce a 2-classifier in prestacks that classifies all discrete opfibrations with small fibres. Finally, we restrict such 2-classifier to a 2-classifier in stacks. This is the main ingredient of a proof that Grothendieck 2-topoi are elementary 2-topoi. Our results also solve a problem posed by Hofmann and Streicher when attempting to lift Grothendieck universes to sheaves. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_16900 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | 2-classifiers via dense generators and Hofmann-Streicher universe in stacks Mesiti, Luca Category Theory Logic 18B25, 18N10, 18F20, 18D30 We expand the theory of 2-classifiers, that are a 2-categorical generalization of subobject classifiers introduced by Weber. The idea is to upgrade monomorphisms to discrete opfibrations. We prove that the conditions of 2-classifier can be checked just on a dense generator. The study of what is classified by a 2-classifier is similarly reduced to a study over the objects that form a dense generator. We then apply our results to the cases of prestacks and stacks, where we can thus look just at the representables. We produce a 2-classifier in prestacks that classifies all discrete opfibrations with small fibres. Finally, we restrict such 2-classifier to a 2-classifier in stacks. This is the main ingredient of a proof that Grothendieck 2-topoi are elementary 2-topoi. Our results also solve a problem posed by Hofmann and Streicher when attempting to lift Grothendieck universes to sheaves. |
| title | 2-classifiers via dense generators and Hofmann-Streicher universe in stacks |
| topic | Category Theory Logic 18B25, 18N10, 18F20, 18D30 |
| url | https://arxiv.org/abs/2401.16900 |