The notion of a (complete) spread was introduced by R.H. Fox [6] in topology in order to give a common generalization of two different types of coverings with singularities (branched and folded). A different notion of a (proper) spread was given by E. Michael [10] in connection with topological cuts. In both cases, the basic ideas is that of a spread, meaning a continuous map p:Y→X, with Y locally connected, satisfying the property that the connected components (or more generally, the clopen subsets) of the p-1(U), for U the opens of X, form a base for the topology of Y.
A topos-theoretic version of the notion of a spread was given by J. Funk and myself [3] as that of a geometric morphism p:F→E between toposes bounded over a base topos S, with F locally connected, for which there is a generating family a:f→p*(E) of F over E which is an S-definable morphism in the sense of Barr and Paré [1]. Thus, we replace "complemented" by "definable", but over a Boolean topos, these two notions agree.
The two types of completions (Fox [6] and Micheal [10]) have a topos-theoretic counterpart, which, unlike the notion of a spread, are far from obvious. We deal with the Fox-like completion in [3] and with the Michael-type completion in [5].
An instance of complete spreads are the branched coverings, introduced in this context in [4] and (with minor variations) in [8]. On the other hand, the notion of folded cover has not been defined for toposes.
Our original interest in complete spreads p:F→E (with a locally connected domain F over S) arose from the correspondence (proven in [3]) with the S-valued Lawvere distributions on E. One direction of the correspondence (the easy direction) had already been noticed by G. Bergman [2] in connection with cosheaves which, as we know, is a notion equivalent to that of distribution and well-known also in topology in connection with homology. In fact, it is clear from [6] that Fox was aware of the correspondence between complete spreads and cosheaves, but did not explicitate.
In this lecture I intend to introduce the notion of a (Fox complete) spread in topology and then in topos theory. This requires a careful study of the classes of definable maps in a topos, as well as an incursion into the localic version of a notion of zero-dimensionality. The notion of "complemented element" can be said in many non-equivalent ways in the localic context (and they all seem to have originated in M. Jibladze, unpublished, but see [9]). The class of complete spreads has some properties (compositionality) but lacks others (unrestricted pullback stability). Its description is (still) too down to earth (meaning that we use sites) and we (Jonathon and I) would like to improve it. Thomas Streicher (private communication) has made some interesting observations of our notion of a spread.
This lecture is based on work done in collaboration with Jonathon Funk [3]. The factorization theorems arising from our analysis in [3], as well as the case of the proper completion (which relates directly to distribution algebras in a fashion which we have not yet managed in the Fox-completion analogue), will prbably require a sequel to this lecture.
References M. Barr and R. Paré, Molecular toposes, J. Pure Appl. Alg. 17(1980) 127-152. G.M. Bergman, Co-rectangular bands and cosheaves in categories of algebras, Alg. Univ. 28(1991) 188-213. M. Bunge and J. Funk, Spreads and the symmetric topos, J. Pure Appl. Alg. 113(1996) 1-38. M. Bunge and S. Niefield, Exponentibility and single universes, J. Pure Appl. Alg. 148(2000) 217-250. M. Bunge, J. Funk, M. Jibladze, T. Streicher, The Michael completion of a spread, to appear in J. Pure Appl. Alg. (Max Kelly volume). R.H. Fox, Covering spaces with singularities, in R.H. Fox et al. (Eds.), Algebraic Geometry and Topology: A Symposium in Honor of S. Lefschetz, Princeton University Press (1957) 243-257. J. Funk, The display locale of a cosheaf, Cahiers de Top. et Géo. Diff. Cat. 36(1995) 53-93. J. Funk, On branced covers in topos theory, Theory and Applications of Categories 7(2000) 1-22. A. Kock and G.E. Reyes, Relatively Boolean and de Morgan toposes and locales, Cahiers de Top. et Géo. Diff. Cat. 35(1994) 249-261. E. Michael, Cuts, Acta Math. 111(1964) 14-30.