  Hochschild lattices are specific intervals in the dexter meet-semilattices
recently introduced by Chapoton. A natural geometric realization of these
lattices leads to some cell complexes introduced by Saneblidze, called the
Hochschild polytopes. We obtain several geometrical properties of the
Hochschild lattices, namely we give cubic realizations, establish that these
lattices are EL-shellable, and show that they are constructible by interval
doubling. We also prove several combinatorial properties as the enumeration of
their $k$-chains and compute their degree polynomials.
