Two cohomology theories for structured spaces
Abstract
In [1] we defined a new kind of space called 'structured space' which locally resembles, near each of its points, some algebraic structure. We noted in the conclusion of the cited paper that the maps fs and h, which are of great importance in the theory of structured spaces, have some connections with the notions of presheaves (and hence also sheaves) and vector bundles. There are well known cohomology theories involving such objects; this suggests the possibility of the existence of (co)homology theories for structured spaces which are somehow related to fs and h. In this paper we indeed develop two cohomology theories for structured spaces: one of them arises from fs, while the other one arises from h. In order to do this, we first develop a more general cohomology theory (called rectangular cohomology in the finite case, and square cohomology in the infinite case), which can actually be applied also in many other situations, and then we obtain the cohomology theories for structured spaces as simple consequences of this theory.