the axioms being the same in both cases. Selecting one or another physical theory amounts to selecting a concrete topos.

In this section we give a topos-theoretic causal theory of space-time. Let $\E$ be an elementary topos with an object of natural numbers, and let $R_{T}$ be the object of continuous real numbers [@St].

[*An affine morphism*]{} $\alpha :R_{T} \to  R_{T}$ is a finite composition of morphisms of the form $$1_{R_{T}}, \quad \otimes  \circ  (\lambda \times 1_{R_{T}}) \circ j ,
\quad \oplus  \circ  (1_{R_{T}}\times \mu ) \circ  j ,$$ where $\oplus ,\otimes $ are the operations of addition and multiplication in $R_{T}$ respectively, $\lambda ,\mu $ are arbitrary elements in $R_{T}$, and $j:R_{T} \simeq 1\times R_{T}$ is an isomorphism. Let $\Gamma $ be the set of all affine morphisms from $R_{T}$ to $R_{T}$.

An [*affine object*]{} in $\E$ is an object $a$ together with two sets of morphisms: $$\Phi  \subset {\rm Hom}_{\E}(R_{T},a), \quad \Psi \subset
{\rm Hom}_{\E}(a, R_{T})$$ such that the following conditions hold:

1\) For any $\phi \in \Phi, \psi \in \Psi $ there is $ \psi \circ \phi \in  \Gamma $.

2\) If $f \in  {\rm Hom} _{\E}(R_{T},a) \setminus \Phi $ then there exists $\psi  \in  \Psi $ 