Geodesic
Bi-directional models of `Radically Synthetic' Differential Geometry
Theory and applications of categories, Bunge Festschrift, Tome 40 (2024), pp. 413-429.

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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.
Publié le :
Classification : 58A03, 18B25, 18F10, 03G30
Keywords: Axiomatic Cohesion, (Radically) Synthetic Differential Geometry 
@article{TAC_2024_40_a14,
     author = {Mat{\'\i}as Menni},
     title = {Bi-directional models of {`Radically} {Synthetic'} {Differential} {Geometry}},
     journal = {Theory and applications of categories},
     pages = {413--429},
     publisher = {mathdoc},
     volume = {40},
     year = {2024},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TAC_2024_40_a14/}
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Matías Menni. Bi-directional models of `Radically Synthetic' Differential Geometry. Theory and applications of categories, Bunge Festschrift, Tome 40 (2024), pp. 413-429. http://geodesic.mathdoc.fr/item/TAC_2024_40_a14/