An Elementary Introduction to n-fold Groupoids

This book introduces groupoids and n -fold groupoids as powerful mathematical tools that unify and generalize local and distant symmetries observed in physical systems. While these concepts are typically discussed in highly specialized literature, this volume makes them accessible to a broader audience by grounding them in concrete applications to solid mechanics, including composite materials and metamaterials. No prior knowledge of the theory or its applications is assumed. The chapters alternate between mathematical foundations and their realization in the behavior of elastic bodies, offering readers a guided and rewarding exploration. Throughout the book, examples and exercises support understanding and encourage hands-on engagement. This work opens a new perspective on symmetry in applied mathematics and physics, bridging abstract theory with practical modeling.

Is the first book devoted to n-fold groupoids The original contextualization within solid mechanics makes abstract mathematical concepts amenable to intuition No previous technical background is needed

Autorentext

Marcelo Epstein is a professor at the University of Calgary, Canada, at the Department of Mechanical and Manufacturing Engineering.

Klappentext

Groupoids and n -fold groupoids unify and generalize the concepts of local symmetries (groups) and distant symmetries of various kinds as they manifest themselves in physical systems. The available mathematical treatments of these topics appear mostly in highly specialized journals and monographs. The aim of this book is to expose the non-specialist to these ideas and to contextualize them within the concrete framework of applications to solid mechanics, such as composite materials and metamaterials. No previous background is assumed in either the theory or the applications. In chapters that alternate the treatment of the mathematical underpinnings with their materialization in the response of elastic bodies the reader is invited to undertake a rewarding journey. Examples and exercises are proposed throughout.

Inhalt

Chapter 1 What is a groupoid?.- Chapter 2 A surgical-mechanical imagery.- Chapter 3 The material groupoid.- Chapter 4 Double groupoids.-Chapter 5 Binary composites.- Chapter 6 Hypercubes abd skeletons.- Chapter 7 Coarse n-groupoids.- Chapter 8 Material n-groupoids.