  A theory of (co)homologies related to set-theoretic $n$-simplex relations is
constructed in analogy with the known quandle and Yang--Baxter (co)homologies,
with emphasis made on the tetrahedron case. In particular, this permits us to
generalize Hietarinta's idea of "permutation-type" solutions to the quantum (or
"tensor") $n$-simplex equations. Explicit examples of solutions to the
tetrahedron equation involving nontrivial cocycles are presented.
