A shelf is a set with a binary operation~\op satisfying a \op (b \op c) =
(a \op b) \op (a \op c). Racks are shelves with invertible translations b
\mapsto a \op b; many of their aspects, including cohomological, are better
understood than those of general shelves. Finite monogenic shelves (FMS), of
which Laver tables and cyclic racks are the most famous examples, form a
remarkably rich family of structures and play an important role in set theory.
We compute the cohomology of FMS with arbitrary coefficients. On the way we
develop general tools for studying the cohomology of shelves. Moreover, inside
any finite shelf we identify a sub-rack which inherits its major
characteristics, including the cohomology. For FMS, these sub-racks are all
cyclic
