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| Format: | Preprint |
| Published: |
2024
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| Online Access: | https://arxiv.org/abs/2405.17748 |
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| _version_ | 1866916262886506496 |
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| author | Menni, Matías |
| author_facet | Menni, Matías |
| contents | The radically synthetic foundation for smooth geometry formulated in [Law11] postulates a space T with the property that it has a unique point and, out of the monoid T^T of endomorphisms, it extracts a submonoid R which, in many cases, is the (commutative) multiplication of a rig structure. The rig R is said to be bi-directional if its subobject of invertible elements has two connected components. In this case, R may be equipped with a pre-order compatible with the rig structure. We adjust the construction of `well-adapted' models of Synthetic Differential Geometry in order to build the first pre-cohesive toposes with a bi-directional R. We also show that, in one of these pre-cohesive variants, the pre-order on R, derived radically synthetically from bi-directionality, coincides with that defined in the original model. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_17748 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Bi-directional models of `radically synthetic' differential geometry Menni, Matías Category Theory 58A03, 18B25, 18F10, 03G30 The radically synthetic foundation for smooth geometry formulated in [Law11] postulates a space T with the property that it has a unique point and, out of the monoid T^T of endomorphisms, it extracts a submonoid R which, in many cases, is the (commutative) multiplication of a rig structure. The rig R is said to be bi-directional if its subobject of invertible elements has two connected components. In this case, R may be equipped with a pre-order compatible with the rig structure. We adjust the construction of `well-adapted' models of Synthetic Differential Geometry in order to build the first pre-cohesive toposes with a bi-directional R. We also show that, in one of these pre-cohesive variants, the pre-order on R, derived radically synthetically from bi-directionality, coincides with that defined in the original model. |
| title | Bi-directional models of `radically synthetic' differential geometry |
| topic | Category Theory 58A03, 18B25, 18F10, 03G30 |
| url | https://arxiv.org/abs/2405.17748 |