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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.}
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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. \\
\begin{theorem}
A geometric morphism (with locally connected domain) is a branched covering iff it satisfies the following three conditions:
\begin{enumerate}[(i)]
\item it is a complete spread
\item it is a purely locally constant covering
\item it is purely skeletal.
\end{enumerate}
\end{theorem}

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

\begin{references}
\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.
\end{references}

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