  We show that the geometry of a Riemannian manifold (M,g) is sensitive to the
apparently purely homotopy-theoretic invariant of M known as the
Lusternik-Schnirelmann category, denoted cat_{LS}(M). Here we introduce a
Riemannian analogue of cat_{LS}(M), called the systolic category of M. It is
denoted cat_{sys}(M), and defined in terms of the existence of systolic
inequalities satisfied by every metric g, as initiated by C. Loewner and later
developed by M. Gromov. We compare the two categories. In all our examples, the
inequality cat_{sys}(M) \le cat_{LS}(M) is satisfied, which typically turns out
to be an equality, e.g. in dimension 3. We show that a number of existing
systolic inequalities can be reinterpreted as special cases of such equality,
and that both categories are sensitive to Massey products. The comparison with
the value of cat(M) leads us to prove or conjecture new systolic inequalities
on M.
