Galois connections and the leray spectral sequence

Abstract

The intent of this paper is to present homological versions of several combinatorial theorems including Rota’s theorem on the Mobius functions of ordered sets joined by a Galois connection and the Crapo complementation theorem. In so doing we strengthen and generalize these results. In addition we make a clear connection between these combinatorial theorems and various forms of the Leray spectral sequence. The basic tool we use for presenting homology is the commutative diagram. As noted in [2], this notion is equivalent to the notion of a sheaf on a suitable topological space associated with the ordered set. In that paper we established that there were significant connections between sheaf cohomology and combinatorial notions such as the Mobius function and Whitney numbers of the first kind. We begin in Section 2 by presenting the terminology of ordered sets and diagrams. The notation we use here is rather more simplified than that of [2]; moreover, we introduce the notion of an “augmented” diagram, which is more appropriate for the combinatorics we present. See also the presentation in [4]. In the next section we develop the concept of a Galois connection on three levels. The first is the usual notion of a Galois correspondence between ordered sets. Next we consider the analogous situation for order-preserving relations. Here we find, in contrast to the case for maps, where Galois correspondences are quite rare, that every order-preserving relation has a Galois adjoint, in fact, that it has many. Finally we consider multirelations. The lack of uniqueness of the Galois adjoint for relations now disappears, and we find that (under a suitable finiteness condition) every multirelation has a unique Galois adjoint. The significance of this adjoint appears in Section 5, where we see that it can be interpreted in terms of the Mobius function of the “fibers” of the orderpreserving relation. .