Spreads and the symmetric topos (Q1815300)

This paper introduces a new and useful tool for the study of the distribution on a topos, of the geometric morphisms and symmetric toposes: the notion of spread for toposes and geometric morphisms. In a diagram \(\begin{matrix}{\mathcal E}&@>\varphi>> &{\mathcal F}\\ \quad e\searrow& &\swarrow f\\& {\mathcal S}\end{matrix}\) in the 2-category of elementary toposes, bounded geometric morphisms and natural transformations, \(\varphi\) is said to be a spread over \(\mathcal S\) if there is a generating family \(E\to \varphi^* F\) for \(\mathcal E\) over \(\mathcal F\) which is \(\mathcal S\)-definable. It is shown that (geometric) spreads are, as in topology, fiberwise zero-dimensional (here zero-dimensionality is defined in terms of relatively complemented opens). In the next section the fiberwise versions of density and purity for geometric morphisms over a base topos are introduced and two factorizations, the pure surjection/spread and the pure dense/complete spread, for geometric morphisms in \(\text{Top}_{\mathcal S}\) with locally connected domain are established. In the process the category of distribution on a topos is characterized in terms of complete spreads. Finally, the above results are applied to the study of the symmetric toposes. In particular it is shown that the symmetric topos is part of a Kock-Zöberlein 2-monad. The authors also give a new construction of bicomma squares in which the ``lower'' leg is an essential geometric morphism (this construction is linked to an interpretation of the equivalence between points of the symmetric topos and complete spreads with locally connected domain as being given by the bicomma square in which the ``lower'' leg is the unit of the symmetric monad). The local connectedness is characterized in terms of the symmetric topos and the bagdomain construction is related via the classifier to a probability distribution.