  We present a research programme aimed at constructing classifying toposes of
Weil-type cohomology theories and associated categories of motives, and
introduce a number of notions and preliminary results already obtained in this
direction. In order to analyze the properties of Weil-type cohomology theories
and their relations, we propose a framework based on atomic two-valued toposes
and homogeneous models. Lastly, we construct a syntactic triangulated category
whose dual maps to the derived categories of all the usual cohomology theories.
