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 and 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 uses an algebraic characterization of homology classes (where is a free abelian covering) which are movable to infinity of with respect to a prescribed cohomology class . This result is established in Part II which can be read independently of the rest of the paper.