the locus stage*]{} $\ell A = \ell C^{\infty}(\R^n)/I \in \L$ of the space-time $R^4$ is the class of $C^{\infty}$-smooth vector functions $(X^0(u), X^1(u),$ $X^2(u),$ $X^3(u)):\R^n\rightarrow  \R^4$, where each function $X^i(u)$ is taken by $mod \ I$,   $I$ is a certain ideal of $C^{\infty}$-smooth functions from $\R^n$ to $\R$. The argument $u\in\R^n$ is some “hidden” parameter corresponding to the stage $\ell A$. Hence it follows that at the stage of real numbers $R=\ell C^{\infty}(\R)$ of the topos under consideration an event $x$ is described by just a $C^{\infty}$-smooth vector function $(X^0(u),X^1(u),X^2(u),X^3(u)), u\in \R$. At the stage of $R^2=\ell C^{\infty}(\R^2)$ an event $x$ is 2-dimensional surface, i.e. a [*string*]{}. The classical four numbers $(x^0,\ x^1,\ x^2,\ x^3)$, the coordinates of the event $x$, are obtained at the stage ${\bf 1}=\ell C^{\infty}(\R^0)= \ell C^{\infty}(\R)/(t)$ (the ideal $(t)$ allows one to identify functions if their values at $0$ coincide), i.e., $x^i=X^i(0), i=0,1,2,3$.

The space-time transformations $f:R^4\rightarrow  R^4$ – are [*elements at the stage $\ell A$*]{} of the functor $$(R^4)^{R^4}\in {\bf Set^{L^{op}}},$$ consisting of the classes of $C^{\infty}$-smooth vector functions $(F^0(u,x),F^1(u,x),$ $ F^2(u,x)$, $F^3(u,x)):\R^n\times\R^4\rightarrow \R^4$, where each function $F^i(u,x)$ is taken by $mod$ of the ideal $\pi^*(I)=(\phi\circ\pi\mid\phi\in I, \pi:\R^{n+4}\rightarrow \R^n
- {\rm projection})$. At the 