Marta Bunge - Covering toposes with singularities

The notion in the title refers to complete spreads in Topos Theory, a generalization due to M. Bunge and J. Funk (1996) of the corresponding topological notion introduced by R. H. Fox (Covering spaces with singularities. In: (eds., R. H. Fox et al), Algebraic Geometry and Topology: a Symposium in Honor of S. Lefschetz, Princeton Univ. Press, 1957, 243-257).

The purpose of this talk is to first review the various equivalent ways to think of complete spreads that are now available, and then to discuss some of the implications of such identifications. For an -bounded topos , the following categories are equivalent to the category of -valued distributions on (in the sense of F. W. Lawvere, Measures in Toposes. Lectures at the Aarhus Workshop in Categorical Methods in Geometry, 1983): (1)  the category of -points of the symmetric topos (M. Bunge, Cosheaves and Distributions on Toposes. Algebra Universalis 34(1995), 469-484); (2)  the category of -complete spreads over with locally connected domains (M. Bunge and J. Funk, Spreads and the Symmetric Topos. J. Pure. Appl. Algebra 113(1996), 1-38); (3)  the category of M-discrete fibrations over in (M. Bunge and J. Funk, On a Bicomma Object condition for KZ-doctrines. J Pure. Appl. Algebra, to appear); and (4)  the dual of the category of distribution algebras in over (M. Bunge, J. Funk and Jibladze, Distribution algebras. in preparation).

Of particular interest in this program is the interplay that takes place between Functional Analysis (Theory of Distributions), Model Theory (Classifying Toposes), Algebraic Topology (Covering Spaces), Category Theory (Fibered Categories) and Lattice Theory (Heyting Algebras).