Galois Theory of Simplicial Complexes

We examine basic notions of categorical Galois theory for the adjunction between Π0 and the inclusion as discrete, in the case of simplicial complexes. Covering morphisms are characterized as the morphisms satisfying the unique simplex lifting property, and are classified by means of the fundamental groupoid, for which we give an explicit “Galoistheoretic” description. The class of covering morphisms is a part of a factorization system similar to the (purely inseparable, separable) factorization system in classical Galois theory, which however fails to be the (monotone, light) factorization. Introduction Out of many good books in algebraic topology, let us pick up R. Brown [4], P. Gabriel and M. Zisman [8], D. Quillen [15], and E. Spanier [17]. We observe: • The Galois theory of covering spaces, i.e. the classification of covering spaces (of a “good” space) via the fundamental groupoid and its actions, is presented in [4] and [8]. It can also be deduced from the results of [17], where however only connected coverings and their canonical projections and automorphisms are considered (rather than the whole category of coverings). • The passage from covering spaces to the actions of the fundamental groupoid, in [4] and [8], uses coverings of groupoids; these are the same as discrete fibrations, called “coverings” by analogy with the topological case. • Gabriel and Zisman [8] also develop what we would call the Galois theory of covering morphisms of simplicial sets. This theory agrees with the topological one via the classical adjunction between geometric realization and the singular complex functor; here again, covering morphisms are defined by analogy with the topological ones. • Spanier [17] does not make use of any combinatorial notion of covering, but constructs directly the so-called edge-path groupoid of a simplicial complex (as recalled in Remark 3.3). * Partially supported by MIUR research projects † Partially supported by MIUR, Australian Research Council and INTAS-97-31961