By Ieke Moerdijk
We strengthen the idea of compactness of maps among toposes, including linked notions of separatedness. This concept is outfitted round types of 'propriety' for topos maps, brought the following in a parallel type. the 1st, giving what we easily name 'proper' maps, is a comparatively susceptible as a result of Johnstone. the second one form of right maps, right here known as 'tidy', fulfill a much better situation because of Tierney and Lindgren.Various different types of the Beck-Chevalley for (lax) fibered product squares of toposes play a valuable function within the improvement of the speculation. purposes contain a model of the Reeb balance theorem for toposes, a characterization of hyperconnected Hausdorff toposes as classifying toposes of compact teams, and of strongly Hausdorff coherent toposes as classifiying toposes of profinite groupoids. Our effects additionally permit us to strengthen extra specific features of the factorization thought of geometric morphisms studied by means of Johnstone. Our ultimate program is a (so-called lax) descent theorem for tidy maps among toposes. This theorem implies the lax descent theorem for coherent toposes, conjectured through Makkai and proved past by way of Zawadowski.