Applied category theory explained

Applied category theory is an academic discipline in which methods from category theory are used to study other fields^{[1] [2] [3]} including but not limited to computer science,^{[4] [5]} physics (in particular quantum mechanics^{[6] [7] [8]}), natural language processing, control theory,^{[9] [10]} probability theory and causality. The application of category theory in these domains can take different forms. In some cases the formalization of the domain into the language of category theory is the goal, the idea here being that this would elucidate the important structure and properties of the domain. In other cases the formalization is used to leverage the power of abstraction in order to prove new results about the field.

List of applied category theorists

• Samson Abramsky
• John C. Baez
  • Bob Coecke Joachim Lambek
• Valeria de Paiva
• Gordon Plotkin
• Dana Scott
• David Spivak

See also

• Categorical quantum mechanics
• ZX-calculus
• DisCoCat
• Petri net
• Univalent foundations
• String diagrams

External links

Journals:

• Compositionality

Conferences:

• Applied category theory
• Symposium on Compositional Structures (SYCO)^[11]

Books:

• Picturing Quantum Processes
• Categories for Quantum Theory
• An Invitation to Applied Category Theory (preprint)
• Category Theory for the Sciences (preprint)

Institutes:

• the Quantum Group at the University of Oxford
• TallCat, a research group at Tallinn University of Technology
• Topos Institute
• Cybercat Institute

Software:

• DisCoPy, a Python toolkit for computing with string diagrams
• CatLab.jl, a framework for applied category theory in the Julia language
• CQL, a query language based on Kan extensions

Companies:

• Conexus AI, a data integration company

Mascots:

• Gremlin-Morgoth

References

1. Web site: Applied Category Theory. MIT OpenCourseWare. 2019-07-20.
2. Book: An Invitation to Applied Category Theory by Brendan Fong. Spivak. David I.. Fong. Brendan. July 2019. 10.1017/9781108668804. 9781108668804. 199139551.
3. Bradley. Tai-Danae. 2018-09-16. What is Applied Category Theory?. math.CT. 1809.05923v2.
4. Book: Barr, Michael.. Michael Barr (mathematician)

   . Category theory for computing science. Michael Barr (mathematician). 1990. Prentice Hall. Wells, Charles.. 0131204866. New York. 19126000.

5. Ehrig. Hartmut. Hartmut Ehrig. Große-Rhode. Martin. Wolter. Uwe. 1998-03-01. Applications of Category Theory to the Area of Algebraic Specification in Computer Science. Applied Categorical Structures. 6. 1. 1–35. 10.1023/A:1008688122154. 290074. 1572-9095.
6. Duncan. Ross. Coecke. Bob. Bob Coecke. 2011. Interacting Quantum Observables: Categorical Algebra and Diagrammatics. New Journal of Physics. 13. 4. 043016. 10.1088/1367-2630/13/4/043016. 0906.4725. 2011NJPh...13d3016C. 14259278.
7. Book: Picturing quantum processes : a first course in quantum theory and diagrammatic reasoning. Coecke. Bob. Bob Coecke. Kissinger. Aleks. 978-1107104228. 1026174191. 2017-03-16.
8. Book: Categories for Quantum Theory: An Introduction. Heunen. Chris. Vicary. Jamie. 9780198739616. 2019-11-19.
9. Master. Jade. Baez. John C.. John Baez. 2018-08-16. Open Petri Nets. math.CT. 1808.05415v4.
10. Baez. John C.. John Baez. Pollard. Blake S.. 2018. A compositional framework for reaction networks. Reviews in Mathematical Physics. 29. 9. 1750028–425. 10.1142/S0129055X17500283. 0129-055X. 2017RvMaP..2950028B. 1704.02051. 119665423.
11. Web site: The n-Category Café. golem.ph.utexas.edu. 2019-07-20.