Ebook: Models for Smooth Infinitesimal Analysis

The aim of this book is to construct categories of spaces which contain all the C?-manifolds, but in addition infinitesimal spaces and arbitrary function spaces. To this end, the techniques of Grothendieck toposes (and the logic inherent to them) are explained at a leisurely pace and applied. By discussing topics such as integration, cohomology and vector bundles in the new context, the adequacy of these new spaces for analysis and geometry will be illustrated and the connection to the classical approach to C?-manifolds will be explained.

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The aim of this book is to construct categories of spaces which contain all the C?-manifolds, but in addition infinitesimal spaces and arbitrary function spaces. To this end, the techniques of Grothendieck toposes (and the logic inherent to them) are explained at a leisurely pace and applied. By discussing topics such as integration, cohomology and vector bundles in the new context, the adequacy of these new spaces for analysis and geometry will be illustrated and the connection to the classical approach to C?-manifolds will be explained.

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The aim of this book is to construct categories of spaces which contain all the C?-manifolds, but in addition infinitesimal spaces and arbitrary function spaces. To this end, the techniques of Grothendieck toposes (and the logic inherent to them) are explained at a leisurely pace and applied. By discussing topics such as integration, cohomology and vector bundles in the new context, the adequacy of these new spaces for analysis and geometry will be illustrated and the connection to the classical approach to C?-manifolds will be explained.
Content:
Front Matter....Pages i-x
Introduction....Pages 1-14
C ?-Rings....Pages 15-55
C ?-Rings as Variable Spaces....Pages 57-96
Two Archimedean Models for Synthetic Calculus....Pages 97-131
Cohomology and Integration....Pages 133-180
Connections on Microlinear Spaces....Pages 181-237
Models with Invertible Infinitesimals....Pages 239-292
Smooth Infinitesimal Analysis....Pages 293-344
Back Matter....Pages 345-399

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The aim of this book is to construct categories of spaces which contain all the C?-manifolds, but in addition infinitesimal spaces and arbitrary function spaces. To this end, the techniques of Grothendieck toposes (and the logic inherent to them) are explained at a leisurely pace and applied. By discussing topics such as integration, cohomology and vector bundles in the new context, the adequacy of these new spaces for analysis and geometry will be illustrated and the connection to the classical approach to C?-manifolds will be explained.
Content:
Front Matter....Pages i-x
Introduction....Pages 1-14
C ?-Rings....Pages 15-55
C ?-Rings as Variable Spaces....Pages 57-96
Two Archimedean Models for Synthetic Calculus....Pages 97-131
Cohomology and Integration....Pages 133-180
Connections on Microlinear Spaces....Pages 181-237
Models with Invertible Infinitesimals....Pages 239-292
Smooth Infinitesimal Analysis....Pages 293-344
Back Matter....Pages 345-399
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