Proper maps of toposes / I. Moerdijk, J.J.C. Vermeulen.

Format:

Book

Author/Creator:

Moerdijk, Ieke, author.

Vermeulen, J. J. C. (Jacob Johan Caspar), 1957- author.

Series:

Memoirs of the American Mathematical Society ; no. 705.

Memoirs of the American Mathematical Society, 0065-9266 ; number 705

Language:

English

Subjects (All):

Toposes.

Mappings (Mathematics).

Physical Description:

1 online resource (125 p.)

Edition:

1st ed.

Place of Publication:

Providence, Rhode Island : American Mathematical Society, [2000]

Language Note:

English

Summary:

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.

Contents:

""Table of Contents""; ""Introduction""; ""Chapter I. Proper Maps""; ""Â1 Definition and Examples""; ""Â2 First Properties""; ""Â3 Beck-Chevalley Conditions""; ""Â4 Pretopos Sites""; ""Â5 Preservation under Pullback and Filtered Inverse Limits""; ""Â6 Propriety and Closed Maps""; ""Â7 Descent along Proper Maps""; ""Chapter II. Separated Maps""; ""Â1 Definition and Examples""; ""Â2 Formal Properties""; ""Â3 Hyperconnected Hausdorff Toposes""; ""Â4 Locally Connected and Locally Compact Maps of Locales""; ""Â5 A Topos Version of the Reeb Stability Theorem""

""Â6 The Classical Reeb Stability Theorem""""Chapter III. Tidy Maps""; ""Â1 Definition and Examples""; ""Â2 First Properties""; ""Â3 The Beck-Chevalley Condition""; ""Â4 Stability under Change of Base""; ""Â5 Entire Maps""; ""Â6 Tidiness and Closed Maps""; ""Chapter IV. Strongly Separated Maps""; ""Â1 Definition of Strong Separation""; ""Â2 Elementary Properties""; ""Â3 Strongly Separated Coherent Toposes""; ""Â4 Galois Theory for Profinite Groupoids""; ""Chapter V. Relatively Tidy Maps and Lax Descent""; ""Â1 Path Toposes""; ""Â2 Lax Pullbacks of Toposes""

""Â3 Relatively Tidy Maps""""Â4 Relatively Tidy Morphisms of Sites""; ""Â5 The Main Theorem""; ""Â6 Applications to Lax Descent""; ""References""

Notes:

Description based upon print version of record.

Includes bibliographical references.

Description based on print version record.

ISBN:

1-4704-0296-3