Sheaf theory through examples / / Daniel Rosiak.

• Superior document: The MIT Press
• VerfasserIn: Rosiak, Daniel,
• Place / Publishing House: Cambridge, Massachusetts : : The MIT Press,, [2022]
• Year of Publication: 2022
• Edition: 1st ed.
• Language: English
• Series: The MIT Press
• Physical Description: 1 online resource (642 pages)

"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"--

Table of 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.