Sheaf theory through examples / Daniel Rosiak.

Format:

Book

Author/Creator:

Rosiak, Daniel, author.

Series:

The MIT Press

Language:

English

Subjects (All):

Sheaf theory.

Physical Description:

1 online resource (642 pages)

Edition:

1st ed.

Place of Publication:

Cambridge, Massachusetts : The MIT Press, [2022]

Language Note:

English

Summary:

"This book presents copious and sometimes unexpected examples of sheaf theory, a mathematical tool with promising applications in data science and engineering and in efforts to apply category theory more widely"-- Provided by publisher.

Contents:

Intro

Title Page

Copyright

Dedication

Contents

Acknowledgments

Introduction

1. Categories

1.1. Categorical Preliminaries

1.2. A Few More Examples

1.3. Returning to the Definition and Distinctions of Size

1.4. Some New Categories from Old

1.5. Aside on "No Objects"

2. Prelude to Sheaves: Presheaves

2.1. Functors

2.2. Natural Transformations

2.3. Seeing Structures as Presheaves

2.4. The Presheaf Action

2.5. Philosophical Pass: The Four Action Perspectives

3. Universal Constructions

3.1. Limits and Colimits

3.2. Philosophical Pass: Universality and Mediation

4. Topology: A First Pass at Space

4.1. Motivation

4.2. A Dialogue Introducing the Key Notions of Topology

4.3. Topology and Topological Spaces More Formally

4.4. Philosophical Pass: Open Questions

5. First Look at Sheaves

5.1. Sheaves: The Topological Definition

5.2. Examples

5.3. Philosophical Pass: Sheaf as Local-Global Passage

6. There's a Yoneda Lemma for That

6.1. First, Enrichment!

6.2. Downsets and Yoneda in the Miniature

6.3. Representability Simplified

6.4. More on Representability, Fixed Points, and a Paradox

6.5. Yoneda in the General

6.6. Philosophical Pass: Yoneda and Relationality

7. Adjunctions

7.1. Adjunctions through Morphology

7.2. Adjunctions through Modalities

7.3. Some Additional Adjunctions and Final Thoughts

7.4. Philosophical Pass: The Idea of Adjointness

8. Sheaves Revisited

8.1. Three Historically Significant Examples

8.2. What Is Not a Sheaf?

8.3. Presheaves and Sheaves in Order Theory

9. Cellular Sheaf Cohomology through Examples

9.1. Simplices and Their Sheaves

9.2. Sheaf Cohomology

9.3. Philosophical Pass: Sheaf Cohomology

9.4. A Glimpse into Cosheaves

10. Sheaves on a Site.

10.1. Revisiting Covers: Toward General Sheaves

10.2. Grothendieck Toposes

10.3. A Few More Examples

10.4. Philosophical Pass: The Idea of Toposes

11. Elementary Toposes

11.1. The Subobject Classifier

11.2. Examples of Elementary Toposes

11.3. Lawvere-Tierney Topologies and Their Sheaves

11.4. Morphisms of Toposes

11.5. Toward Cohesive Toposes

A: Appendix (Revisiting Topology)

A.1. Conceptual Motivation: Topology as Logic of Finite Observations

A.2. Explicit Connections to Modal Logic

A.3. The Idea of All This

A.4. Why Opens?

A.5. What Is Topology Really About?

References

Index.

Notes:

OCLC-licensed vendor bibliographic record.

ISBN:

0-262-36237-6

0-262-37042-5

OCLC:

1333708310