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"-- Provided by publisher.

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.