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.

一本易於理解的初級層狀理論介紹書，並探討其在純數學以外的應用。

層狀理論是一種數學構造，關注從局部性質到全局性質的轉換。它在現代數學的許多領域中扮演了基礎性的角色，然而層狀理論的廣泛概念力和其在純數學以外領域的廣泛應用僅在最近才開始受到重視。以應用範疇論的觀點，「通過例子學習層狀理論」提供了一個易於理解的初級層狀理論介紹，並探討了包括圖形的n-著色、衛星數據、棋盤問題、貝葉斯網絡、自相似群、音樂表演、複合體等應用。

本書強調通過豐富的有動機和生動插圖的例子來發展理論，並在拓撲學、範疇論和層狀理論的哲學反思中補充了概念的正式發展，同時選擇了一些高級主題和例子，以說明細胞層狀上同調、拓撲學、幾何擬態等思想。

「通過例子學習層狀理論」旨在將數學家使用的層狀理論的強大結果與現實世界的應用相結合，同時通過獨特的哲學觀點補充技術問題，以適應更廣泛的思想發展。

Daniel Rosiak is a Research Associate at the Inamori International Center for Ethics and Excellence at Case Western Reserve University.

Daniel Rosiak 是凱斯西儲大學伊那莫里國際倫理與卓越中心的研究助理。