  In this paper we study a new notion of category weight of homology classes
developing further the ideas of
  E. Fadell and S. Husseini. In the case of closed smooth manifolds the
homological category weight is equivalent to the cohomological category weight
of E. Fadell and S. Husseini but these two notions are distinct already for
Poincar\'e complexes. An important advantage of the homological category weight
is its homotopy invariance. We use the notion of homological category weight to
study various generalizations of the Lusternik - Schnirelmann category which
appeared in the theory of closed one-forms and have applications in dynamics.
Our primary goal is to compare two such invariants $\cat(X,\xi)$ and
$\cat^1(X,\xi)$ which are defined similarly with reversion of the order of
quantifiers. We compute these invariants explicitly for products of surfaces
and show that they may differ by an arbitrarily large quantity. The proof of
one of our main results, Theorem \ref{main2}, uses an algebraic
characterization of homology classes $z\in H_i(\tilde X;\Z)$ (where $\tilde
X\to X$ is a free abelian covering) which are movable to infinity of $\tilde X$
with respect to a prescribed cohomology class $\xi\in H^1(X;\R)$. This result
is established in Part II which can be read independently of the rest of the
paper.
