awesome-category-theory

In this course we explain how category theory has become useful for writing elegant and maintainable code. In particular, we'll use examples from the Haskell programming language to motivate category-theoretic constructs. By Brendan Fong, Bartosz Milewski, and David Spivak (2020)

David Spivak and Brendan Fong

Lecture Notes from University of Munich by Bodo Pareigis (2004)

This lecture is part of an online course on commutative algebra, following the book "Commutative algebra with a view toward algebraic geometry" by David Eisenbud. By Richard E. Borcherds (2020).

Lecture Notes from Stanford University (2022)

Connects 3-crossed modules with quasi-categories through the Moore complex (2025)

This paper presents a comprehensive graphical theory for discrete linear time-invariant systems, expanding on classical signal flow diagrams to handle streams with infinite pasts and futures, introduces a new structural view on controllability, and is grounded in the extended theory of props by Brendan Fong, Paolo Rapisarda and Paweł Sobociński (2015)

The paper introduces the Giry monad as a categorical tool for defining and studying random processes and related concepts by Michèle Giry (2016)

Bayesian inference and decision making on measurable spaces with countably generated σ-algebras, using regular conditional probabilities and Eilenberg--Moore algebras by Jared Culbertson, Kirk Sturtz (2013)

Drawing from Spivak, Fong, and Cruttwell et al.'s foundational works, this study establishes a categorical framework for Bayesian inference, incorporating concepts of Bayesian inversions by Kotaro Kamiya, John Welliaveetil (2021)

Introduces channels in a graphical language to define and study conjugate priors in Bayesian probability, and shows how they ensure the same class of distributions for prior and posterior by Bart Jacobs (2017)

Comprehensive yet approachable, this exceptional book covers all subjects addressed in a standard introductory abstract algebra course

Explores advanced subjects in the realm of universal algebra. The core content is organized into two chapters, each addressing different aspects of universal algebra within the framework of category theory. The first chapter introduces the concept of triplable categories, inspired by the theory of modules over a ring, and explores the equivalence between categories of triples in any given category and theories over that category. In the second chapter, Davis shifts focus to equational systems of functors, a more generalized approach to algebra that encompasses both the triplable and structure category theories. Dissertation by Robert Clay Davis (1967)

Covers topics such as the fundamentals of algebraic theories, free models, special theories, the completeness of algebraic categories, and extends to more complex concepts like commutative theories, free theories, and the Kronecker product, among others. The notes also touch on the rings-theories analogy proposed by F. W. Lawvere, suggesting an insightful correlation between rings/modules and algebraic theories/models. Gavin C. Wraith (1975)

Discusses the concept of valued functors in category theory, particularly focusing on tensor products. Freyd explores the application of algebraic theories in non-standard categories, starting with the question of what constitutes an algebra in the category of sets, using category predicates without elements. The text outlines the axioms of a group using category theory language, emphasizing objects and maps. Peter Freyd (1966)

Explores the field of universal algebra with a particular focus on the concept of algebras over a triple. The work is grounded in the realization that categories of algebras, traditionally defined with finitary operations and satisfying a set of equations, can be extended to include infinitary operations as well, thereby broadening the scope of universal algebra. Manes starts by discussing the conventional understanding of universal algebra, tracing back to G.D. Birkhoff's definition in the 1930s, and then moves to explore how this definition can be expanded by considering sets with infinitary operations. Dissertation by Ernest Gene Manes (1967)

Discusses the embedding of a small category A into the category of contravariant functors from A to Set (the category of sets), which preserves inverse limits but does not generally preserve direct limits. Kock introduces a "codensity monad" for any functor from a small category to a left complete category and explores the universal generator for this monad. He demonstrates that the Yoneda embedding followed by this generator provides a full and faithful embedding that is both left and right continuous. Additionally, the relationship with Isbell's adjoint conjugation functors and the definition of generalized (direct and inverse) limit functors are addressed, by Anders Kock (1966).

Explores the concept of exact categories and the theory of derived functors, building upon earlier work by Buchsbaum. Freyd investigates how properties and statements applicable to abelian groups can extend to arbitrary exact categories. Freyd aims to formalize this observation into a metatheorem, which would simplify categorical proofs and predict lemmas. Peter J. Freyd's dissertation, presented at Princeton University (1960)

Applied Category Theory Conference

This book offers a beginner-friendly introduction to category theory, a versatile conceptual framework used across various disciplines, detailing fundamental concepts, examples, and over 200 exercises, making it ideal for self-study or as a course text, by Harold Simmons (2011)

Tom Leinster's (2014) book represents an edited version of his lecture notes. As such, it is a concise work that provides focused coverage of the Category Theory topics it addresses

Monoidal category theory provides an abstract language to describe quantum theory, emphasizing intuitive graphical calculus, and explores structures modeling quantum phenomena, classical information, and probabilistic systems, with connections to other disciplines highlighted throughout by Chris Heunen, Jamie Vicary (2020)

The content is in-depth, and its mathematical aspects can be challenging for the reader. It's advisable to explore this book after reading one or two of the more introductionary books. This book is a classic by Saunders Mac Lane (1971)

This book offers an in-depth yet accessible introduction to category theory, targeting a diverse audience and covering essential concepts; the second edition includes expanded content, new sections, and additional exercises by Steve Awodey (2010)

This book introduces Category Theory at a level appropriate for computer scientists and provides practical examples (in Haskell) in the context of programming languages by Bartosz Milewski (2019)

Series of lectures based on the "Seven Sketches" book - by Brendan Fong and David I. Spivak

Steve Awodey has an excellent series, aimed a little higher (with a compsci flavour), going a little further

Material will be presented at a tutorial level that will help graduate students and researchers understand the critical issues and open problems confronting the field. By Steve Awodey (2012)

This is a video-based course aimed at post-graduate students and as well academics interested to learn about category theory, with live participation of the audience shaping the content of the course. (2022)

A Discord Server about Applied Category Theory by the Topos Institute

Category Theory Channel on the biggest Discord Math Server