$.

An element $u \in  p_{x}$ is called [*extreme*]{} if there exists $\phi  \in  \Phi_{0} $ for which $u \in \phi $, but $ y \notin  \phi $ for all $y \in  p_{x}, y \neq u$.

A conic order $\O$ is said to be [*strict*]{} if, for each nonextreme element $u \in p_{x}$, and $v \in p_x, v \neq u$, and each ray $\lambda$ with origin $u$ such that $v \in \lambda $, there exists an extreme element $w \in \lambda $, and $w \in p_x$.

2.2. An affine object $a$ with an order $\O$, which is complete, strict, conic, has an acute vertex, and is invariant with respect to $\A$ is said to be [*Lorentz*]{} if for each $x:1\to a$ and each extreme elements $u, v \in p_{x}$, where $u, v\neq x$ there exists a $f \in Aut(\O)$ such that $ f \circ u = v, f \circ x = x $.

2.1. A Lorentz object in the topos [**Set**]{} is an affine space admitting a pseudo-Euclidean structure defined by a quadratic form $$x_{0}^{2} - \sum_{i=1}^{n}x_{i}^{2},$$ where $n$ is finite or equal to $\infty $, and $ Aut(\O)$ is the Poincaré group (see [[@AGS]]{}). A Lorentz object in the 