  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.
