Arrows, structures, and functors : the categorical imperative / Michael A. Arbib, Ernest G. Manes

This book attempts to build up sufficient perspective on category theory without demanding more of the reader than a basic knowledge of sets and matrix theory. Category theory is a mathematician's attempt to lay bare some of the underlying principles common to diverse fields in the mathematical sciences. It has become, as well, an area of pure mathematics in its own right. The contribution of our book, then, is to build up a sufficient perspective without demanding more of the reader than a basic knowledge of sets (what is a function? what is a Cartesian product of sets?) and matrix theory (what is a linear map? what is a direct product of vector spaces?) In short, this is by far the most elementary category theory text in print. The et of core concepts of category theory are First is the ability to think with to express key concepts in terms of mappings rather than in terms of set elements. Second is the realization that collections of mathematical structures find convenient characterization in terms of arrows. Third is the use of functors as the appropriate tools with which to compare different domains of mathematical discourse.The book is divided into two parts. The first part (Chapters 1 to6) is devoted to arrows and structures, The second part (Chapters 7 to 10) is an introduction to functors. Contents Preface Part I 1 Learning to Think with Arrows 2 Basic Concepts of Category Theory 3 Monoids and Groups 4 Metric and Topological Spaces 5 Additive Categories 6 Structured Sets Part II 7 Functors and Adjoints 8 The Adjoint Functor Theorem 9 Monoidal and Closed Categories 10 Monads and Algebras Suggestions for Further Reading Index Michael A. Arbib, Ernest G. Manes. Includes Index. Bibliography: P. 176-177.