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Ø[[O]{}]{} \#1\_\#2\^\#3

**Toposes in General Theory\
of Relativity\
**

Alexandr K. Guts,   Egor B. Grinkevich

Department of Mathematics, Omsk State University\
644077 Omsk-77 RUSSIA\
Email: guts@univer.omsk.su, grinkev@univer.omsk.su\
October 30, 1996\
ABSTRACT

We study in this paper different topos-theoretical approaches to the problem of construction of General Theory of Relativity. In general case the resulting space-time theory will be non-classical, different from that of the usual Einstein theory of space-time. This is a new theory of space-time, created in a purely logical manner. Four possibitities are investigated: axiomatic approach to causal theory of space-time, the smooth toposes as a models of Theory of Relativity, Synthetic Theory of Relativity, and space-time as Grothendieck topos.

Introduction
============

Construction of a causal theory of space-time is one of the most attractive tasks of science in the 20th century. From the viewpoint of mathematics, partially ordered structures should be considered. The latter is commonly understood as a set $M$ with a specified reflexive and transitive binary relation $\preceq$. A primary notion is actually not that of causality but rather that of motion (interaction) of material objects. Causality is brought to the foreground since an observer detects changes 