Sheaf Theory through Examples

An approachable introduction to elementary sheaf theory and its applications beyond pure math.

Sheaves are mathematical constructions concerned with passages from local properties to global ones. They have played a fundamental role in the development of many areas of modern mathematics, yet the broad conceptual power of sheaf theory and its wide applicability to areas beyond pure math have only recently begun to be appreciated. Taking an applied category theory perspective, Sheaf Theory through Examples provides an approachable introduction to elementary sheaf theory and examines applications including n-colorings of graphs, satellite data, chess problems, Bayesian networks, self-similar groups, musical performance, complexes, and much more.
 
With an emphasis on developing the theory via a wealth of well-motivated and vividly illustrated examples, Sheaf Theory through Examples supplements the formal development of concepts with philosophical reflections on topology, category theory, and sheaf theory, alongside a selection of advanced topics and examples that illustrate ideas like cellular sheaf cohomology, toposes, and geometric morphisms.
 
Sheaf Theory through Examples seeks to bridge the powerful results of sheaf theory as used by mathematicians and real-world applications, while also supplementing the technical matters with a unique philosophical perspective attuned to the broader development of ideas.Acknowledgments ix
Introduction 1
1 Categories 21
2 Prelude to Sheaves: Presheaves 45
3 Universal Constructions 89
4 Topology: A First Pass at Space 109
5 First Look at Sheaves 147
6 There's a Yoneda Lemma for That 171
7 Adjunctions 195
8 Sheaves Revisited 241
9 Cellular Sheaf Cohomology through Examples 277
10 Sheaves on a Site 311
11 Elementary Toposes 369
A Appendix (Revisiting Topology) 411
References 431
Index 439"Remarkably approachable for those wanting to learn about the theory of sheaves but who do not have an extensive familiarity with the deep abstraction common to mathematical theories in general."
—M. Clay, University of Arkansas“Rosiak stimulates a shift in perspective—one that will encourage readers from diverse backgrounds to learn sheaf theory and imagine new applications to science, engineering and philosophy.”
—Justin Curry, Professor of Mathematics and Statistics at SUNY Albany

“Rosiak's visual imagination and his very detailed treatment of a wide array of examples allow the reader to uncover, as Novalis' Novices of Sais, the secrets of sheaf theory, perhaps the deepest mathematical construction of the twentieth century.”
—Fernando Zalamea, Professor of Mathematics, Universidad Nacional de Colombia

“With precise definitions and brimming with creative and illustrative examples, ranging from the Penrose Triangle to image processing, Sheaf Theory through Examples is probably the most pleasant way to learn about sheaves.”
—David Spivak, Chief Scientist, Topos InstituteDaniel Rosiak is a Research Associate at the Inamori International Center for Ethics and Excellence at Case Western Reserve University.