Hinze R. Introducing String Diagrams. The Art of Category Theory 2023

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Hinze R. Introducing String Diagrams. The Art of Category Theory 2023

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Textbook in PDF format String diagrams are powerful graphical methods for reasoning in elementary category theory. Written in an informal expository style, this book provides a self-contained introduction to these diagrammatic techniques, ideal for graduate students and researchers. Much of the book is devoted to worked examples highlighting how best to use string diagrams to solve realistic problems in elementary category theory. A range of topics are explored from the perspective of string diagrams, including adjunctions, monad and comonads, Kleisli and Eilenberg–Moore categories, and endofunctor algebras and coalgebras. Careful attention is paid throughout to exploit the freedom of the graphical notation to draw diagrams that aid understanding and subsequent calculations. Each chapter contains plentiful exercises of varying levels of difficulty, suitable for self-study or for use by instructors. Prologue Category Theory Categories Properties of Arrows Thinking in Terms of Arrows Functors Natural Transformations Properties of Functors Equational Reasoning Exercises String Diagrams Composition of Functors Composition of Natural Transformations Converting between Symbols and Diagrams Equational Reasoning Natural Isomorphisms Duality Exercises Monads Monads Monad Maps Comonads Kleisli Categories Eilenberg-Moore Categories Actions of a Monad Exercises Adjunctions Adjunction Reasoning with Adjunctions Composition of Adjunctions Mates Adjoint Comonads and Monads Reflective and Coreflective Subcategories Equivalences of Categories Exercises Putting It All Together Huber's Construction Universal Constructions Free Monads The Resumption Monad Exercises Epilogue Appendix Notation References Index