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Macquarie University  Department of Mathematics

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Categories in Algebra, Geometry and Mathematical Physics

Conference in honour of Ross Street's sixtieth birthday

July 11-16 2005, Macquarie University, Sydney

Fri, 15 July:  14:00 - 14:40

An intrinsic characterization of branched coverings
Bunge, Marta (McGill University, Montreal)

\thanks{This is joint work with Jonathan Funk.}
In my lecture I will present a result within the rich theory of singular maps of toposes known as complete spreads~\bibref{1}. A geometric morphism with locally connected domain is a spread if it is zero-dimensional, and it is complete if cogerms of components converge. There is an equivalence~\bibref{2} between complete spreads over any $S$-bounded topos and Lawvere $S$-valued distributions, for any base topos $S$. This equivalence is part of an adjunction of the comprehensive type which is analogous to the comprehensive factorization of Street--Walters~\bibref{3}, but which is relative to the symmetric monad~\bibref{4}.

In topology, complete spreads were introduced in~\bibref{5} in order to deal with branched coverings in a topological rather than a combinatorial manner. A geometric morphism over a topos $E$ (with locally connected domain) is a branched covering if it is the spread completion of a locally constant covering of a slice topos $E/U$ for some pure subobject $U$ of $1$ in $E$. This definition --- given independently in~\bibref{6} and~\bibref{7}, is inspired by that given in~\bibref{5} for topological spaces.

We now seek to give an intrinsic characterization of branched coverings as a subclass of the complete spreads. To this end, we introduce some new notions in what follows. A monomorphism in a topos is said to be {\em pure} if the constant object $2$ has the sheaf property with respect to every pullback of the monomorphism. We call {\em purely skeletal} any geometric morphism whose inverse image functor preserves pure monomorphisms. The notion of a {\em purely locally constant covering} is obtained from that of a locally constant covering~\bibref{8} by allowing the splitting object to have a pure (but not necessarily global) support. \\
A geometric morphism (with locally connected domain) is a branched covering iff it satisfies the following three conditions:
\item it is a complete spread
\item it is a purely locally constant covering
\item it is purely skeletal.

Examples showing the independence between the three notions appearing in the characterization above will be given.

\bibitem M. Bunge and J. Funk, {\em Toposes, distributions, and complete spread maps}, Book. Submitted February 2005.
\bibitem M. Bunge and J. Funk, {\em Spreads and the symmetric topos}, J. Pure Appl. Alg. 113 (1996) 1--28.
\bibitem R. Street and R.F.C. Walters, {\em The comprehensive factorization of a functor}, Bull. Amer. Math. Soc. 79 (1973) 936--941.
\bibitem M.Bunge and A. Carboni, {\em The symmetric topos}, J. Pure Appl. Alg. 105 (1995) 233--249.
\bibitem R. H. Fox, {\em Covering spaces with singularities}, in Algebraic Geometry and Topology : A Symposium in Honor of S. Lefschetz, Princeton University Press, 1957, 243--257.
\bibitem M. Bunge and S. B. Niefield, {\em Exponentiability and single universes}, J. Pure Appl. Alg. 148 (2000) 217--250.
\bibitem J. Funk, {\em On branched covers in topos theory}, Theory and Applications of Categories 7 (2000) 1--22.
\bibitem M. Bunge and S. Lack, {\em Van Kampen theorems in toposes}, Advances in Mathematics 179 (2003) 291--317.

Typeset PDF of this abstract.