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Proper Maps of Toposes
 
I. Moerdijk Utrecht University, Utrecht, Netherlands
J. J. C. Vermeulen University of Cape Town, Rondebusch, South Africa
Proper Maps of Toposes
eBook ISBN:  978-1-4704-0296-9
Product Code:  MEMO/148/705.E
List Price: $52.00
MAA Member Price: $46.80
AMS Member Price: $31.20
Proper Maps of Toposes
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Proper Maps of Toposes
I. Moerdijk Utrecht University, Utrecht, Netherlands
J. J. C. Vermeulen University of Cape Town, Rondebusch, South Africa
eBook ISBN:  978-1-4704-0296-9
Product Code:  MEMO/148/705.E
List Price: $52.00
MAA Member Price: $46.80
AMS Member Price: $31.20
  • Book Details
     
     
    Memoirs of the American Mathematical Society
    Volume: 1482000; 108 pp
    MSC: Primary 18; 22;

    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.

    Readership

    Graduate students and research mathematicians interested in category theory, homological algebra.

  • Table of Contents
     
     
    • Chapters
    • Introduction
    • I. Proper maps
    • II. Separated maps
    • III. Tidy maps
    • IV. Strongly separated maps
    • V. Relatively tidy maps and lax descent
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 1482000; 108 pp
MSC: Primary 18; 22;

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.

Readership

Graduate students and research mathematicians interested in category theory, homological algebra.

  • Chapters
  • Introduction
  • I. Proper maps
  • II. Separated maps
  • III. Tidy maps
  • IV. Strongly separated maps
  • V. Relatively tidy maps and lax descent
Review Copy – for publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Please select which format for which you are requesting permissions.