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By Ieke Moerdijk

We increase 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 right here in a parallel style. the 1st, giving what we easily name 'proper' maps, is a comparatively vulnerable situation as a result of Johnstone. the second one type of right maps, the following referred to as 'tidy', fulfill a better situation because of Tierney and Lindgren.Various sorts of the Beck-Chevalley for (lax) fibered product squares of toposes play a significant function within the improvement of the speculation. functions comprise 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 allow us to boost extra specific points of the factorization concept of geometric morphisms studied via 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 prior by means of Zawadowski.

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