  A construction of hexagon relations - algebraic realizations of
four-dimensional Pachner moves - is proposed. It goes in terms of "permitted
colorings" of 3-faces of pentachora (4-simplices), and its main feature is that
the set of permitted colorings is nonconstant - varies from pentachoron to
pentachoron. Further, a cohomology theory is formulated for these hexagon
relations, and its nontriviality is demonstrated on explicit examples.
