Net bundles over posets and K-theory

We proceed with the study of net bundles over partially ordered sets (posets), defined as the analogues of ordinary fibre bundles. To this end, we analyze the connection between homotopy, net homology and net cohomology of a poset, providing versions of classical Hurewicz theorems. Focusing our attention to Hilbert net bundles, we define the K-theory of a poset and introduce functions over the homotopy groupoid satisfying the same formal properties of Chern classes. In the case in which the given poset is a subbase for the topology of a space, our results describe the category of locally constant bundles.