Sheaf theory through examples

Sheaf theory through examples

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"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"--

Detalles Bibliográficos
Otros Autores: Rosiak, Daniel, author (author)
Formato: Libro electrónico
Idioma:Inglés
Publicado: Cambridge, Massachusetts : The MIT Press [2022]
Edición:1st ed
Colección:The MIT Press
Materias:
Sheaf theory.
MATHEMATICS / Logic
MATHEMATICS / Topology
MATHEMATICS / Algebra / General
Ver en Biblioteca Universitat Ramon Llull:https://discovery.url.edu/permalink/34CSUC_URL/1im36ta/alma991009701153806719
Tabla de Contenidos:
  • 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.

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