Bi-directional models of `radically synthetic' differential geometry

TL;DR

This paper develops bi-directional models within radically synthetic differential geometry, constructing pre-cohesive toposes where the algebraic structure R exhibits bi-directionality and compatible pre-orders, advancing the foundational understanding of synthetic geometry.

Contribution

It introduces the first pre-cohesive toposes with bi-directional rig structures R, extending synthetic differential geometry to new algebraic and categorical frameworks.

Findings

Constructed pre-cohesive toposes with bi-directional R.

Demonstrated the coincidence of pre-orders in a specific model.

Extended synthetic differential geometry to bi-directional algebraic structures.

Abstract

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.