Lusternik-schnirelmann Cocategory

We introduce a new deenition of the (Lusternik-Schnirelmann) co-category of a CW complex X. This is accomplished by producing a dual of the fat wedge called the thin product. One then looks at fac-torizations of the fold map through the thin product analogously to the Whitehead deenition of category. We investigate the relationship between this deenition and the rational cocategory introduced by Sba. We provide a counterexample to Sba 's conjecture that rational co-category and the localization of Ganea's inductive cocategory are the same.