From old categorical models to new ones through open covers

Abstract

The nerve theorem provides a combinatorial model for the homotopy type of a topological space from a suitable open cover. We extend this local-to-global approach by replacing the intersections of the cover with compatible categorical models. These local models are assembled through the Grothendieck construction, yielding a small category whose classifying space has the weak homotopy type of the original space. Under suitable hypotheses, the resulting category is finite and acyclic. We develop two versions of this construction. The first is indexed by the usual poset of nonempty finite intersections of the cover. The second uses the membership poset, inspired by the finite-space construction of Sancho de Salas, which retains only the membership patterns realized by points of the space and is defined for point-finite covers. The classical nerve theorem and its componentwise variant are recovered as particular cases. The construction is recursive: categorical models of local pieces, together with functors representing their inclusions, can be combined to produce categorical models of more complex spaces.