Proper maps of toposes
About this Title
I. Moerdijk and J. J. C. Vermeulen
Publication: Memoirs of the American Mathematical Society
Publication Year 2000: Volume 148, Number 705
ISBNs: 978-0-8218-2168-8 (print); 978-1-4704-0296-9 (online)
MathSciNet review: 1787303
MSC: Primary 18B25; Secondary 18F10, 22A22
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.
Graduate students and research mathematicians interested in category theory, homological algebra.
Table of Contents
- I. Proper maps
- II. Separated maps
- III. Tidy maps
- IV. Strongly separated maps
- V. Relatively tidy maps and lax descent