Reference. The graphical theory of monads
The formal theory of monads shows that much of the theory of monads can be developed in the abstract at the level of 2-categories. This means that results about monads can be established once and for all and simply instantiated in settings such as enriched category theory. Unfortunately, these results can be hard to reason about as they involve more abstract machinery. In this paper, we present the formal theory of monads in terms of string diagrams — a graphical language for 2-categorical calculations. Using this perspective, we show that many aspects of the theory of monads, such as the Eilenberg–Moore and Kleisli resolutions of monads, liftings, and distributive laws, can be understood in terms of systematic graphical calculational reasoning. This paper will serve as an introduction both to the formal theory of monads and to the use of string diagrams, in particular, their application to calculations in monad theory.
Cite
Cites 50 works (4 here)
With notes (4)
Introducing String Diagrams: The Art of Category Theory hinze-2023-introducing
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.
Compositional Game Theory ghani-2018-compositional
Equational reasoning with lollipops, forks, cups, caps, snakes, and speedometers hinze-2016-equational
Kan Extensions for Program Optimisation Or: Art and Dan Explain an Old Trick hinze-2012-kan
External (46)
- Exploring String Diagrams - The Art of Category Theory (2025)
- Diagrammatic Algebra of First Order Logic (2024)
- The formal theory of relative monads (2024)
- Quantum in Pictures (2022)
- Regular calculi I: Graphical regular logic (2021)
- Backprop as Functor: A compositional perspective on supervised learning (2019)
- Categories for Quantum Theory (2019)
- Graphical linear algebra (2019)
- A compositional treatment of iterated open games (2018)
- Picturing Quantum Processes: A First Course in Quantum Theory and Diagrammatic Reasoning (2017)
- Categorical semantics of digital circuits (2016)
- A compositional framework for Markov processes (2016)
- Dragging Proofs Out of Pictures (2016)
- String diagrams for free monads (functional pearl) (2016)
- String diagrams for double categories and equipments (2016)
- Monads need not be endofunctors (2015)
- Categories in control (2015)
- Full abstraction for signal flow graphs (2015)
- A compositional framework for passive linear networks (2015)
- Category Theory using String Diagrams (2014)
- A Tannakian context for Galois theory (2013)
- Iterated distributive laws (2011)
- A Survey of Graphical Languages for Monoidal Categories (2010)
- Mathematical foundations for distributed compositional models of meaning (2010)
- A 2-Categories Companion (2009)
- A unified framework for generalized multicategories (2009)
- The Joy of String Diagrams (2008)
- The formal theory of monads II (2002)
- Categories for the Working Mathematician (1998)
- Categorical Structures (1996)
- Higher categories, strings, cubes and simplex equations (1995)
- Categories, allegories and circuit design (1994)
- The geometry of tensor calculus, I (1991)
- Making formality work for us (1989)
- Planar diagrams and tensor algebra (1988)
- On the Shape of Mathematical Arguments (1988)
- String diagrams for non-abelian cocycle conditions (1987)
- Review of the elements of 2-categories (1974)
- The formal theory of monads (1972)
- Distributive laws (1969)
- Continuous Yoneda representation of a small category (1966)
- Every standard construction is induced by a pair of adjoint functors (1965)
- Adjoint functors and triples (1965)
- Catégories structurées (1963)
- Homotopy theory in general categories (1961)
- Adjoint functors (1958)