Covering morphisms in topos theory
by
Marta Bunge
Department of Mathematics and Statistics, McGill University, Montreal, Canada

In the work of Janelidze in 1990, a formal notion of covering morphism arises from an abstract categorical framework given by a pair of adjoint functors. Associated with any such class of covering morphisms, there is a corresponding (pure) Galois theory, of which there are examples in different areas of mathematics. In topos theory, the class of all covering projections (local homeomorphisms determined by a locally constant object of the topos) that appears explicitly in the work of Barr and Diaconescu in 1981, has been shown already by Janelidze to be an instance of the above notion of covering morphism, but only under the conditions that either the base topos be Set (that is, for Grothendieck toposes), or else where the splitting cover in the topos is assumed connected.

In 1991, and independently of the above, I gave a categorical construction of the fundamental groupoid of a locally connected topos E, bounded over an arbitrary base topos S via a geometric morphism e \colon E --> S, from which all of the relevant properties were easily derived. In this construction, a different notion of locally constant object emerged that seemed to be at the same time more general as well as more appropriate than that of Barr and Diaconescu, even though both were meaningful in the same context.

I briefly recall the definition of the fundamental groupoid of E (relative to S) and thus of the origin of this more general notion. For each cover U\twoheadrightarrow 1 in E, a topos G_U(E) is defined by means of the pushout, in the 2-category Top_S of bounded S-toposes and where e! \dashv e^* \dashv e_* is part of the data for the locally connected geometric morphism e \colon E --> S, of E/U --> E and E/U --> S/e!U, where the latter is the connected and locally connected portion of the unique factorization of the composite E/U --> E --> S into a connected locally connected followed by a local homeomorphism. A suitable limit topos, taken over a small cofinal system of covers, is then shown, using descent in two different forms, to be the classifying topos B(G), for G an etale complete prodiscrete localid groupoid in S which represents first-degree cohomolgy of E with coefficients in discrete groups, and which then is defined as the fundamental (localic) groupoid of E. A comparison with the paths version of the fundamental group of a connected locally connected topos, constructed as topos by Moerdijk-Wraith (1986), was established in Bunge-Moerdijk (1997).

The topos G_U(E), as a category, is equivalent to the full subcategory of E determined by its ``locally constant objects'', but in the following sense (Bunge-Lack 2001). An object A of E is said to be locally constant, split by a cover U \twoheadrightarrow 1 in E, if there exists a morphism J --> I in S, a morphism U --> e^*I in E, and an isomorphism

U ×A --> U ×e*I e*(J)

over U.

This says precisely that the projection U ×A --> U is S-definable in the sense of Barr and Pare, that is, it says that A --> 1 is U-locally S-definable. In turn, this says that U ×A --> U is ``trivial'' (and so A --> 1 is locally trivial) with respect to the adjunction e! \dashv e^*, in the sense of Janelidze. Note that for these equivalences, no special assumptions are needed on either the base topos S or on the cover U. [The realization of the above identifications between these notions of locally constant and locally trivial came to me during an illuminating discussion with George Janelidze in Sydney in November 2001.]

In my lecture at the workshop, I will first discuss the connection between my work nd that of Janelidze, in particular the various equivalences between notions of locally constant, and how the fundamental groupoid of a (locally simply connected) topos appears then as the Galois groupoid of the adjunction e! \vdash e^*. Another aspect that I will discuss in this connection is that of a (relative) notion of Galois topos, previously considered (by Grothendieck in 1972, and by Moerdijk in 1998) only in the case of pointed Grothendieck toposes. In addition, I will also discuss another possible instance of the of the covering projections, namely the class of all unramified complete spread coverings (Bunge-Funk 1996, 1998). This example is potentially of interest, not only because of its connection with the Lawvere distributions, but also because it seems to fit into a Magid-Janelidze framework involving locally connected toposes instead of commutative rings, and distribution algebras (Bunge-Funk-Jibladze-Streicher 2000) instead of Boolean algebras.

Date received: May 20, 2002

Copyright © 2002 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cajf-21.