The generalized Lusternik-Schnirelmann category of a product space

We continue to study the notions of Y-category and strong Vcategory which we introduced in [2]. We give a characterization of them in terms of homotopy colimits and then use it to prove some product theorems in this context. In [2] we introduced new homotopy invariants by generalizing the notions of Lusternik-Schnirelmann category and strong category as follows: For any given class of spaces v we define the X'-category X-cat(X) of a space X to be the smallest integer k for which there exists a numerable covering (X1, ..., Xk) of X such that each inclusion X. c X factors through some space Ai E v up to homotopy. If no such covering exists we set X-cat(X) oo. X-cat(X) is an invariant of the homotopy type of X [2, 1.4]. If we replace the condition that each X c X factors through some space in v up to homotopy by asking each X to have the homotopy type of some space in X, then the number V-gcat(X) thus obtained is not a homotopy invariant [3]. So we define the strong X-category sl-Cat(X) of X to be the minimum of Ygcat(X') for all spaces X' having the homotopy type of X. If v consists only of the one-point-space then X-cat (X) is just LusternikSchnirelmann's category cat (X) of X and V-Cat (X) is the strong category Cat (X) introduced by Ganea in [5]. Other interesting examples are the qconnective (strong) category catq(X) (Catq(X)) of X obtained by taking -v to be the class of q-connected CW-complexes, q > 0, and the q-dimensional (strong) category catq(X) (Catq (X)) of X obtained by taking for -v the class of all CW-complexes of dimension < q [2, 1.2]. In this paper we shall prove the following product theorems, which generalize the well known ones for the classical case [7]. Unlike [7] however, we do not restrict ourselves to polyhedra but rather exploit the linear structure given by the numeration of the coverings. Received by the editors November 11, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 55M30; Secondary 55P50.