  We study probabilistic variants of the Lusternik--Schnirelmann category and
topological complexity, which bound the classical invariants from below. We
present a number of computations illustrating both wide agreement and wide
disagreement with the classical notions. In the aspherical case, where our
invariants are group invariants, we establish a counterpart of the
Eilenberg--Ganea theorem in the torsion-free case, as well as a contrasting
universal upper bound in the finite case.
