Memoirs of the American Mathematical Society 2000; 108 pp; softcover Volume: 148 ISBN10: 0821821687 ISBN13: 9780821821688 List Price: US$49 Individual Members: US$29.40 Institutional Members: US$39.20 Order Code: MEMO/148/705
 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 BeckChevalley 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 (socalled 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. Readership Graduate students and research mathematicians interested in category theory, homological algebra. Table of Contents  Introduction
 Proper maps
 Separated maps
 Tidy maps
 Strongly separated maps
 Relatively tidy maps and lax descent
 References
