On this page I have collected a few selected resources about what (applied) category theory is, how to learn more about it, and the roles it is playing in diverse fields across science and engineering.

Category theory is a branch of mathematics that traditionally has not been taught in science and engineering. Reasons for this include its (erroneous) reputation as being “too abstract” to be useful for applications, and the fact that, historically, older subjects such as linear algebra, calculus, and probability theory are emphasized in university curricula.

However, applications of category theory are well-established in physics and computer science, and over the past several decades it has increasingly become part of cutting-edge research in engineering and applied sciences, in such diverse domains as robotics, epidemiology, natural language processing, or agent-based modeling.

Short video: "What is category theory?" by Paul Dancstep
A first primer for a general audience.

Blog post: What is Category Theory, Anyway?, by Tai-Danae Bradley.
A short description of category theory, via some key ideas and mathematical notions and examples. Assumes some mathematical knowledge, and provides many useful links for further reading.

Notes: What is Applied Category Theory?, by Tai-Danae Bradley.
A friendly introduction to some ideas, mathematical concepts, and examples of applied category theory. Assumes some technical mathematical background, and is sprinkled with links and references for further study.

Book chapter on category theory, written by Spencer Breiner and Eswaran Subrahmanian, in the Handbook of Model-Based Systems Engineering.
"Category theory (CT) is a branch of mathematics concerned with the representation and composition of structured relationships. Recent interest in systems engineering (SE) stems from the possibility that CT might provide a principled mathematical foundation that SE currently lacks. The case is bolstered by a broad array of existing applications in probability, computing, data and dynamics, as well as a track record of unification in science and mathematics. However, the tools and methodology for applying CT within engineering are mostly at the level of prototype and proof-of-concept, and there is significant research needed to adapt these methods to an SE context."

Brief article by Ilyas Khan, founder of Quantinuum.
This post provides an informative short overview of some of the history of category theory, and then focuses on more specialized recent developments in one of the many areas where category theory is being applied: quantum computing.

Talk by Eugenia Cheng: The joy of abstract mathematics
An accessible presentation for a general audience and which explains, in particular, how "abstract" can also be very down to earth.

Talk by Ken Scambler introducing applied category theory
A presentation to a computer science oriented audience which describes some broad recent developments in applied category theory and explains a number of key concepts and ideas.

Talk by Kris Brown: Scientific and software engineering examples of applied category theory
"Abstraction is something all programmers are familiar with. We like to add abstractions in the form of little helpful scripts and functions in order to save time and avoid repeating ourselves; however, these eventually start to not fit well together, become hard to modify, and, most importantly, cannot be made sense of by others. Ad hoc abstractions are stacked like a house of cards; eventually it's easier to start over from scratch once our scientific models or workflows have to update. Category theory (CT) is math suitable for talking about abstraction, and thinking about our scientific models and workflows with CT leads to abstractions which fit well together."

Book: The Joy of Abstraction (2022), by Eugenia Cheng.
A very accessible, conceptually illuminating, and technically rigorous introduction to the mathematics of category theory. Written for an audience with no previous math knowledge. There was a book club where Eugenia answered readers’ questions about each chapter, and see also this author interview.

Book: Seven Sketches in Compositionality: An Invitation to Applied Category Theory (2018), by Brendan Fong and David Spivak.
A wonderful introduction to applied category theory, accessible to readers with basic mathematical knowledge at an undergraduate level. Also see John Baez’s series of online written lectures, based on this book, and the videos from the MIT course taught by Fong and Spivak.

Book: Category Theory for the Sciences (2014), by David Spivak.
Probably the first ever textbook specifically for applied category theory in the sciences. Accessible to readers with mathematical knowledge at an undergraduate level.

Article: Physics, Topology, Logic and Computation: A Rosetta Stone (2009), by John C. Baez and Mike Stay.
A famous and illuminating paper that showed one important way that category theory links four different disciplines.

Research

Since it's birth in the 1940's, category theory has produced an enormous wealth of research. Nowadays, researcher in category theory is mainly undertaken by academics in mathematics, computer science, and physics departments, as well as researchers in indpendent institutes and industry. In the following we highlight just a few sources that are particularly relevant for applied category theory.

• The Topos Institute is a hub of leading research in applied category theory and hosts a regular colloquium, organizes workshops and seminars, develops software, and posts regularly on their blog.
• Compositionality is a journal specifically founded to publish research in applied category theory and compositional structures more generally.
• Proceedings of the annual conference in applied category theory (published on the arXiv) are a useful place to get a sense of research in applied category theory. Once on a specific proceedings arXiv page, click on “HTML” to view a table of contents with links.