Sheaf Theory through Examples

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