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List: awesome-applied-ct

applied-category-theory category-theory compositionality curated-list references

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# Awesome Applied Category Theory



A curated list of Applied Category Theory resources.



## Table of contents



- [Awesome ACT](#awesome-act)

    - [Articles](#articles)

        - [Databases](#databases)

        - [Data types](#data-types)

        - [Dynamical systems](#dynamical-systems)

        - [Game theory](#game-theory)

        - [Machine learning](#machine-learning)

        - [Petri nets](#petri-nets)

    - [Books](#books)

    - [Companies](#companies)

        - [Heavily using ACT](#heavily-using-act)

        - [Interested in ACT research](#interested-in-act-research)

    - [Conferences](#conferences)

    - [Journals](#journals)

    - [Lectures](#lectures)

    - [Meetups](#meetups)

    - [Software libraries](#software-libraries)

    - [Tools](#tools)



### Articles

*Relevant articles applying category theory to various fields.*



#### Databases



* [Algebraic databases](http://www.tac.mta.ca/tac/volumes/32/16/32-16abs.html) - by [Patrick Schultz][schultz], [David I. Spivak][spivak], [Christina Vasilakopoulou][vasilakopoulou] and [Ryan Wisnesky][wisnesky]

* [Algebraic Model Management: A survey](https://www.categoricaldata.net/cql/wadt.pdf) - by [Patrick Schultz][schultz], [David I. Spivak][spivak], and [Ryan Wisnesky][wisnesky]

* [Functorial data migration](https://www.sciencedirect.com/science/article/pii/S0890540112001010) - by [David I. Spivak][spivak]



#### Data types



* [Categories of Containers](https://www.cs.nott.ac.uk/~psztxa/publ/fossacs03.pdf) - by [Michael Abbot][abbot], [Thorsten Altenkirch][altenkirch] and [Neil Ghani][ghani]



#### Dynamical systems



* [A categorical approach to open and interconnected dynamical systems](https://arxiv.org/abs/1510.05076) - by [Brendan Fong][fong], [Paolo Rapisarda][rapisarda] and [Paweł Sobociński][sobocinski]



#### Game theory



* [A semantical approach to equilibria and rationality](https://arxiv.org/abs/0905.3548) - by [Dusko Pavlovic][pavlovic]

* [Compositional game theory](https://arxiv.org/abs/1603.04641) - by [Jules Hedges][hedges], [Neil Ghani][ghani], [Viktor Winschel][winschel] and [Philipp Zahn][zahn]

* [The game semantics of game theory](https://arxiv.org/abs/1904.11287) - by [Jules Hedges][hedges]



#### Linguistics



* [Free compact 2-categories](https://hal-lirmm.ccsd.cnrs.fr/lirmm-00137681v2/document) - by Joachim Lambek and Anne Preller

* [Mathematical foundations for a compositional distributional model of meaning](https://arxiv.org/abs/1003.4394) - by [Bob Coecke][coecke], [Mehrnoosh Sadrzadeh][sadrzadeh] and [Stephen Clark][clark]

* [The Frobenius anatomy of word meanings I: subject and object relative pronouns](https://arxiv.org/abs/1404.5278) - by [Mehrnoosh Sadrzadeh][sadrzadeh], [Stephen Clark][clark] and [Bob Coecke][coecke]



#### Machine learning



* [Backprop as Functor: A compositional perspective on supervised learning](https://arxiv.org/abs/1711.10455) - by [Brendan Fong][fong], [David I. Spivak][spivak] and [Rémy Tuyéras][tuyeras]

* [Compositional Deep Learning](https://arxiv.org/abs/1907.08292) - by [Bruno Gavranović][gavranovic]



#### Manufacturing



* [String diagrams for assembly planning](https://arxiv.org/abs/1909.10475) - by [Jade Master][master], [Evan Patterson][patterson], Shahin Yousfi, Arquimedes Canedo



#### Petri nets



* [Generalized Petri Nets](https://arxiv.org/abs/1904.09091) - by [Jade Master][master]

* [The Mathematical Specification of the Statebox Language](https://arxiv.org/abs/1906.07629) - by [Fabrizio Genovese][genovese] and [Jelle Herold][herold]



#### Probability and statistics



* [A categorical approach to probability theory](https://www.chrisstucchio.com/blog_media/2016/probability_the_monad/categorical_probability_giry.pdf) - by Michèle Giry

* [Causal inference by string diagram surgery](https://arxiv.org/abs/1811.08338) - by [Bart Jacobs][jacobs], [Aleks Kissinger][kissinger] and [Fabio Zanasi][zanasi]

* [A synthetic approach to Markov kernels, conditional independence and theorems on sufficient statistics](https://arxiv.org/abs/1908.07021) - by [Tobias Fritz][fritz]



### Books

*Books on applied category theory.*



* [Category Theory for Programmers](https://github.com/hmemcpy/milewski-ctfp-pdf) - by [Bartosz Milewski][milewski]

* [Category Theory for the Sciences](https://mitpress.mit.edu/books/category-theory-sciences) - by [David I. Spivak][spivak]

* [Seven Sketches in Compositionality: An Invitation to Applied Category Theory](https://arxiv.org/abs/1803.05316) - by [Brendan Fong][fong] and [David I. Spivak][spivak] (note also the [corresponding lecture series](#lectures)).

* [What is Applied Category Theory](https://arxiv.org/abs/1809.05923) - by [Tai-Danae Bradley][bradley]



### Companies

*Companies applying category theory.*



#### Heavily using ACT



* [Conexus](https://conexus.com/) - A start-up developing [CQL](https://www.categoricaldata.net/), 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](https://statebox.org/) - building a formally verified process language using robust mathematical principles to prevent errors, allow compositionality and ensure termination



#### Interested in ACT research



* [IOHK](https://iohk.io/) - builds cryptocurrencies and blockchain solutions, based on peer reviewed papers; formally verified specifications in [Agda](https://github.com/input-output-hk/plutus/tree/master/metatheory), [Coq](https://github.com/input-output-hk?language=coq) and [k-framework](https://testnet.iohkdev.io/iele/about/formal-verification/)

* [RChain](https://github.com/rchain/) - blockchain ecosystem it's foundational language - Rholang is implementation of [rho-calculus](http://rho.loria.fr/index.html) wih deep roots in [higher category theory](https://arxiv.org/abs/1504.04311) and [enriched Lawvere theories](https://arxiv.org/abs/1704.03080)



### Conferences



* [ACT](http://www.appliedcategorytheory.org/) - Applied Category Theory Conference

* [Statebox Summit](https://summit.statebox.org/) - An yearly gathering of category theorists and functional programmers

* [SYCO](http://events.cs.bham.ac.uk/syco) - Symposium on Compositional Structures



### Journals

*Academic journals on applied category theory.*



* [Compositionality](http://www.compositionality-journal.org/) - 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](https://www.youtube.com/playlist?list=PLhgq-BqyZ7i5lOqOqqRiS0U5SwTmPpHQ5), series of lectures based on the "Seven Sketches" book - by [Brendan Fong][fong] and [David I. Spivak][spivak]



### Meetups

* [Boston](https://www.meetup.com/Categorical-Databases/)

* [Munich](https://www.meetup.com/Applied-Category-Theory-Munich/)

* [New York](https://www.meetup.com/NYC-Category-Theory/)

* [San Francisco Bay Area](https://www.meetup.com/Category-Theory)



### Software libraries

*Libraries to actually encode category theory in software.*



* [copumpkin/categories](https://github.com/copumpkin/categories) - categories parametrized by morphism equality, in [Agda](https://wiki.portal.chalmers.se/agda/pmwiki.php)

* [idris-ct](https://github.com/statebox/idris-ct) - a formally verified category theory library written in [Idris](https://www.idris-lang.org/)

* [UniMath](https://github.com/UniMath/UniMath/tree/master/UniMath/CategoryTheory) - categories formalized using univalent mathematics, in [Coq](https://coq.inria.fr/)



### Tools

*Useful tools for applied category theory.*



* [Cartographer.id](http://cartographer.id/) - a tool for string diagrammatic reasoning

* [Catlab.jl](https://github.com/epatters/Catlab.jl) - an experimental framework for applied category theory

* [Homotopy.io](https://homotopy.io/) - a web-based proof assistant for finitely-presented globular n-categories

* [KdMonCat](https://edit.statebox.cloud/) - a tool for drawing morphisms in monoidal categories



[abbot]: https://www.cs.le.ac.uk/people/mabbott/

[altenkirch]: http://www.cs.nott.ac.uk/~psztxa/

[bradley]: https://www.math3ma.com/

[clark]: https://sites.google.com/site/stephenclark609/

[coecke]: https://www.cs.ox.ac.uk/people/bob.coecke/

[fong]: http://brendanfong.com/

[fritz]: https://perimeterinstitute.ca/personal/tfritz/

[gavranovic]: https://www.brunogavranovic.com/

[genovese]: https://twitter.com/fabgenovese

[ghani]: https://www.strath.ac.uk/staff/ghanineilprof/

[hedges]: https://julesh.com/

[herold]: http://shell.defekt.nl/~jelle/#/p/about/

[jacobs]: https://www.cs.ru.nl/B.Jacobs/

[kissinger]: https://www.cs.ru.nl/A.Kissinger/papers.html

[master]: https://sites.google.com/view/jadeedenstarmaster/home

[milewski]: https://bartoszmilewski.com/

[patterson]: https://www.epatters.org/

[pavlovic]: http://dusko.org/

[rapisarda]: https://www.ecs.soton.ac.uk/people/pr3

[sadrzadeh]: https://msadrzadeh.com/

[schultz]: https://www.linkedin.com/in/patrick-schultz-16645590/

[sobocinski]: https://www.ioc.ee/~pawel/

[spivak]: https://math.mit.edu/~dspivak/

[tuyeras]: http://www.normalesup.org/~tuyeras/

[vasilakopoulou]: https://mathdept.ucr.edu/faculty/cvasil.html

[winschel]: http://www.vikwin.de/

[wisnesky]: https://www.wisnesky.net/

[zahn]: https://www.philipp-zahn.com/

[zanasi]: http://www.zanasi.com/fabio/