Covering Theory and the fundamental progroupoid of a general topos

EDUARDO J. DUBUC

University of British Columbia, Vancouver, Canada

Congreso; International Conference in Category Theory CT2004; 2004

Pacific Institute of Mathematical Science

COVERING THEORY AND THE FUNDAMENTAL PROGROUPOID OF A GENERAL TOPOS The galois theory of locally constant objects for a locally connected topos is well developed and understood, and as it is, it depends heavily on the local condition of connectedness. Recently topologist working in shape theory have dealt with the theory for a non locally connected base space. In their work, the descent data underneath the notion of covering projection has to be made explicit in one way or another. We realized that we are in face of a situation of classical topological descent as described in the introduction to ``Categories Fibrees et Descente´´, Expose VI, SGA1. We approach the notion of locally constant object X trivialized by a cover U focusing in the descent data that construct X. We introduce a condition on this data that it is vacuous when the topos is locally connected, and that when the topos is spatial, it furnish by descent the notion of covering projection on non locally connected base spaces considered by topologists. Given an arbritrary topos E we define the topos G_U of covering projections split by U to be the category of locally constant objects constructed by descent data that satisfies the condition. The key result is that this topos is locally connected even when E is not. Theorem: The topos G_U is an atomic topos with a canonical surjective point f_U. Theorem: There is an equivalence between G_U and the topos B{P_U}, where P_U is the localic groupoid of automorphisms of the point f_U, and B{P_U} its classifying topos. The assignments of P_U and G_U to U are functorial on the bifiltered category of coverings under refinement, and by restriction to the filtering poset of covering sieves define the fundamental progroupoid and the fundamental protopos of E. The fundamental topos is the inverse limit topos corresponding to the fundamental protopos. It comes equipped with a localic point defined over the inverse limit of the sets of objects of the grupoids P_U. The theorems above hold for these inverse limit topoi.