to the topos [**Set**]{}. This gives rise to the topos ${\bf Set^P}$, and it is this topos which is the [*new mathematical space-time*]{}.

The value of a functor $F$ on an element $x$ of ${\bf P}$ is the set $F(x)$. The set ${\bf P}$ is interpreted as the collection of all possible situations of obtaining information about past. It has a temporal partial order. The set $F(x)$ is the (causal) past cone consisting of the events that are observed in situation $x$. The functor $F$ can be interpreted as a time flow. The topos ${\bf Set^P}$ consists of all possible time flows. It is not hard to see that a classical Lorentz transformation corresponds to a natural isomorphism of functors, i.e. time flows. In fact, consider two different time flows $F$ and $G$, and let $x,y, \ \  x \preceq y$ are two time situations. Then we have the following diagram:

$$\begin{aligned}
     x \quad    & \preceq         &  \quad y    \\
                &                 &                \\
      F(x)      & \stackrel{F(\preceq)}{\longrightarrow} & F(y) \\
\downarrow \tau_x  &          & \downarrow \tau_y   \\
      G(x)& \stackrel{G(\preceq) }{\longrightarrow} & G(y)\end{aligned}$$

where $\tau$ is a natural isomorphism of functors $F, G$. Let all $F(x), G(x), \ x\in {\bf P}$ are 