topos ${\bf Top}(M)$ is a fiber bundle over $M$ with fibers equipped with an affine structure and a continuous pseudo-Euclidean structure of finite or infinite dimension.

It is quite possible to take not only the topoi ${\bf Set}$, ${\bf Bn}(M)$, or ${\bf Top}(M)$, but also any others that have an affine object.

The existing categorical determination of the set theory and determination of ${\bf Top}(M)$ between elementary topoi gives the possibility to speak about the solution of problem of categorical description of the Theory of Relativity. 2.2. If $\E$ is a well-pointed topos satisfying the axiom of partial transitivity with a Lorentz object $a$, then $\E$ is a model of set theory $ Z$ and $a$ is a model of the Special Relativity. If $\E$ is a topos defined over [**Set**]{} that has enough points and satisfies the axiom (SG) (see [[@Jo]]{}) with a Lorentz object $a$, then $\E$ is a topos ${\bf Top}(M)$ and $a$ is a model of the General Relativity.

Smooth toposes as a models of Theory of Relativity
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There exists a simple way to construct a non-classical Theory of Relativity with intuitionistic logic by using the objects of smooth toposes $${\bf Set^{L^{op}}, \ {\bf Sh(L)}, 