#
Bi-directional models of `Radically Synthetic' Differential Geometry

##
Matías Menni

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.

Keywords: Axiomatic Cohesion, (Radically) Synthetic Differential Geometry

2020 MSC:
58A03, 18B25, 18F10, 03G30

*Theory and Applications of Categories,*
Vol. 40, 2024,
No. 15, pp 413-429.

Published 2024-05-23.

http://www.tac.mta.ca/tac/volumes/40/15/40-15.pdf

TAC Home