Internal 1-topoi in 2-topoi
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910544909303808 |
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| author | Helfer, Joseph |
| author_facet | Helfer, Joseph |
| contents | We further develop Weber's notion of elementary 2-topos by proposing certain new axioms. We show that in a 2-category C satisfying these axioms, the "discrete opfibration (DOF) classifier" S is always an internal elementary 1-topos, in an appropriate sense. The axioms introduced for this purpose are closure conditions on the DOFs having "S-small fibres". Among these closure conditions, the most interesting one asserts that a certain DOF, analogous to the "subset fibration" over Set, has small fibres.
The remaining new axioms concern "groupoidal" objects in a 2-category, which are seen to play a significant role in the general theory. We prove two results to the effect that a 2-category C satisfying these axioms is "determined by" its groupoidal objects: the first shows that C is equivalent to a 2-category of internal categories built out of groupoidal objects, and the second shows that the groupoidal objects are dense in C. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_19358 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Internal 1-topoi in 2-topoi Helfer, Joseph Category Theory Logic 18B25 (Primary) 03G30, 18N10, 18N25 (Secondary) We further develop Weber's notion of elementary 2-topos by proposing certain new axioms. We show that in a 2-category C satisfying these axioms, the "discrete opfibration (DOF) classifier" S is always an internal elementary 1-topos, in an appropriate sense. The axioms introduced for this purpose are closure conditions on the DOFs having "S-small fibres". Among these closure conditions, the most interesting one asserts that a certain DOF, analogous to the "subset fibration" over Set, has small fibres. The remaining new axioms concern "groupoidal" objects in a 2-category, which are seen to play a significant role in the general theory. We prove two results to the effect that a 2-category C satisfying these axioms is "determined by" its groupoidal objects: the first shows that C is equivalent to a 2-category of internal categories built out of groupoidal objects, and the second shows that the groupoidal objects are dense in C. |
| title | Internal 1-topoi in 2-topoi |
| topic | Category Theory Logic 18B25 (Primary) 03G30, 18N10, 18N25 (Secondary) |
| url | https://arxiv.org/abs/2407.19358 |