# Proper Maps of Toposes

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*I. Moerdijk; J. J. C. Vermeulen*

We develop the theory of compactness of maps between toposes,
together with associated notions of separatedness. This theory is built around
two versions of “propriety” for topos maps, introduced here in a
parallel fashion. The first, giving what we simply call “proper”
maps, is a relatively weak condition due to Johnstone. The second kind of
proper maps, here called “tidy”, satisfy a stronger condition due
to Tierney and Lindgren. Various forms of the Beck-Chevalley condition for
(lax) fibered product squares of toposes play a central role in the development
of the theory.

Applications include a version of the Reeb stability theorem
for toposes, a characterization of hyperconnected Hausdorff toposes as
classifying toposes of compact groups, and of strongly Hausdorff coherent
toposes as classifiying toposes of profinite groupoids. Our results also enable
us to develop further particular aspects of the factorization theory of
geometric morphisms studied by Johnstone.

Our final application is a (so-called lax) descent theorem for
tidy maps between toposes. This theorem implies the lax descent theorem for
coherent toposes, conjectured by Makkai and proved earlier by Zawadowski.