Awesome Applied Category Theory

A curated list of Applied Category Theory resources.

Table of contents

• Awesome ACT
  • Articles
    • Databases
    • Data types
    • Dynamical systems
    • Game theory
    • Machine learning
    • Petri nets
  • Books
  • Companies
    • Heavily using ACT
    • Interested in ACT research
  • Conferences
  • Journals
  • Lectures
  • Meetups
  • Software libraries
  • Tools

Articles

Relevant articles applying category theory to various fields.

Databases

• Algebraic databases - by Patrick Schultz, David I. Spivak, Christina Vasilakopoulou and Ryan Wisnesky
• Algebraic Model Management: A survey - by Patrick Schultz, David I. Spivak, and Ryan Wisnesky
• Functorial data migration - by David I. Spivak

Data types

• Categories of Containers - by Michael Abbot, Thorsten Altenkirch and Neil Ghani

Dynamical systems

• A categorical approach to open and interconnected dynamical systems - by Brendan Fong, Paolo Rapisarda and Paweł Sobociński

Game theory

• A semantical approach to equilibria and rationality - by Dusko Pavlovic
• Compositional game theory - by Jules Hedges, Neil Ghani, Viktor Winschel and Philipp Zahn
• The game semantics of game theory - by Jules Hedges

Linguistics

• Free compact 2-categories - by Joachim Lambek and Anne Preller
• Mathematical foundations for a compositional distributional model of meaning - by Bob Coecke, Mehrnoosh Sadrzadeh and Stephen Clark
• The Frobenius anatomy of word meanings I: subject and object relative pronouns - by Mehrnoosh Sadrzadeh, Stephen Clark and Bob Coecke

Machine learning

• Backprop as Functor: A compositional perspective on supervised learning - by Brendan Fong, David I. Spivak and Rémy Tuyéras
• Compositional Deep Learning - by Bruno Gavranović

Manufacturing

• String diagrams for assembly planning - by Jade Master, Evan Patterson, Shahin Yousfi, Arquimedes Canedo

Petri nets

• Generalized Petri Nets - by Jade Master
• The Mathematical Specification of the Statebox Language - by Fabrizio Genovese and Jelle Herold

Probability and statistics

• A categorical approach to probability theory - by Michèle Giry
• Causal inference by string diagram surgery - by Bart Jacobs, Aleks Kissinger and Fabio Zanasi
• A synthetic approach to Markov kernels, conditional independence and theorems on sufficient statistics - by Tobias Fritz

Books

Books on applied category theory.

• Category Theory for Programmers - by Bartosz Milewski
• Category Theory for the Sciences - by David I. Spivak
• Seven Sketches in Compositionality: An Invitation to Applied Category Theory - by Brendan Fong and David I. Spivak (note also the corresponding lecture series).
• What is Applied Category Theory - by Tai-Danae Bradley

Companies

Companies applying category theory.

Heavily using ACT

• Conexus - A start-up developing CQL, a generalization of SQL to data migration and integration that contains an automated theorem prover to rule out most semantic errors at compile time.
• Statebox - building a formally verified process language using robust mathematical principles to prevent errors, allow compositionality and ensure termination

Interested in ACT research

• IOHK - builds cryptocurrencies and blockchain solutions, based on peer reviewed papers; formally verified specifications in Agda, Coq and k-framework
• RChain - blockchain ecosystem it’s foundational language - Rholang is implementation of rho-calculus wih deep roots in higher category theory and enriched Lawvere theories

Conferences

• ACT - Applied Category Theory Conference
• Statebox Summit - An yearly gathering of category theorists and functional programmers
• SYCO - Symposium on Compositional Structures

Journals

Academic journals on applied category theory.

• Compositionality - open-access journal for research using compositional ideas, most notably of a category-theoretic origin, in any discipline

Lectures

Video recordings of lecture series on applied category theory.

• Applied Category Theory @ MIT 2019, series of lectures based on the “Seven Sketches” book - by Brendan Fong and David I. Spivak

Meetups

• Boston
• Munich
• New York
• San Francisco Bay Area

Software libraries

Libraries to actually encode category theory in software.

• copumpkin/categories - categories parametrized by morphism equality, in Agda
• idris-ct - a formally verified category theory library written in Idris
• UniMath - categories formalized using univalent mathematics, in Coq

Tools

Useful tools for applied category theory.

• Cartographer.id - a tool for string diagrammatic reasoning
• Catlab.jl - an experimental framework for applied category theory
• Homotopy.io - a web-based proof assistant for finitely-presented globular n-categories
• KdMonCat - a tool for drawing morphisms in monoidal categories