Fundamental progroupoid and bundles with a structural category

In this paper, for a given space X , a structural category C , and a faithful functor η from C to the category of spaces, we introduce a notion of ( C , η)-bundle which contains as particular cases, the notions of covering space, of overlaying space (introduced by Fox), of suspension foliation and other well-known topological structures. The new notion allows us to use sheaf theory and category theory in order to obtain some classification theorems which appear in terms of equivalences of categories. We prove that the category ( C , η)-bundle(X) of ( C , η)-bundles over X is equivalent to the category pro(πCX, C ) , which is determined by the fundamental groupoid of X and the structural category C . As particular cases we obtain the standard classification of covering spaces, Fox's classification theorem for overlays with a finite number of leaves and the standard classification of suspension foliations. This paper illustrates the importance of the fundamental progroupoid of a space X , which plays in shape theory the role of the standard fundamental groupoid. If the space X satisfies some additional properties, the progroupoid πCX can be reduced to a surjective progroup, a groupoid or a group. In some cases a surjective progroupoid can be replaced by a topological prodiscrete group. In all these cases the category pro(πCX, C ) also reduces to well-known categories.