Applied category theory

Short description: Applications of category theory

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][9]), natural language processing,[10][11][12] control theory,[13][14][15][16] 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 or to develop new algorithms about the field.

## Contents

* 1 List of applied category theorists
* 2 See also
* 3 External links
* 4 References

## 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)[17]

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
* UDBMS, a research group at University of Helsinki

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
* Symbolica, a machine learning company

Mascots:

* Gremlin-Morgoth

## References

1. ↑ "Applied Category Theory". https://ocw.mit.edu/courses/mathematics/18-s097-applied-category-theory-january-iap-2019/.
2. ↑ Spivak, David I.; Fong, Brendan (July 2019). An Invitation to Applied Category Theory by Brendan Fong. doi:10.1017/9781108668804. ISBN 978-1-108-66880-4.
3. ↑ Bradley, Tai-Danae (2018-09-16). "What is Applied Category Theory?". arXiv:1809.05923v2 [math.CT].
4. ↑ Barr, Michael. (1990). Category theory for computing science. Wells, Charles.. New York: Prentice Hall. ISBN 0-13-120486-6. OCLC 19126000.
5. ↑ Ehrig, Hartmut; 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. doi:10.1023/A:1008688122154. ISSN 1572-9095.
6. ↑ Abramsky, Samson; Coecke, Bob (2009), "Categorical Quantum Mechanics", Handbook of Quantum Logic and Quantum Structures (Elsevier): pp. 261–323, doi:10.1016/b978-0-444-52869-8.50010-4, ISBN 978-0-444-52869-8
7. ↑ Duncan, Ross; Coecke, Bob (2011). "Interacting Quantum Observables: Categorical Algebra and Diagrammatics". New Journal of Physics 13 (4). doi:10.1088/1367-2630/13/4/043016. Bibcode: 2011NJPh...13d3016C.
8. ↑ Coecke, Bob; Kissinger, Aleks (2017-03-16). Picturing quantum processes: a first course in quantum theory and diagrammatic reasoning. ISBN 978-1-107-10422-8. OCLC 1026174191.
9. ↑ Heunen, Chris; Vicary, Jamie (2019-11-19). Categories for Quantum Theory: An Introduction. Oxford University Press. ISBN 978-0-19-873961-6.
10. ↑ Coecke, Bob; Sadrzadeh, Mehrnoosh; Clark, Stephen (2011), Mathematical Foundations for a Compositional Distributional Model of Meaning
11. ↑ Kartsaklis, Dimitri; Sadrzadeh, Mehrnoosh; Pulman, Stephen; Coecke, Bob (2016), "Reasoning about meaning in natural language with compact closed categories and Frobenius algebras", Logic and Algebraic Structures in Quantum Computing, Cambridge University Press, pp. 199–222, doi:10.1017/cbo9781139519687.011, ISBN 978-1-139-51968-7
12. ↑ Grefenstette, Edward; Sadrzadeh, Mehrnoosh; Clark, Stephen; Coecke, Bob; Pulman, Stephen (2014), "Concrete Sentence Spaces for Compositional Distributional Models of Meaning", Text, Speech and Language Technology (Springer Netherlands): pp. 71–86, doi:10.1007/978-94-007-7284-7_5, ISBN 978-94-007-7283-0
13. ↑ Bonchi, Filippo; Sobocinski, Pawel; Zanasi, Fabio (2021), "A Survey of Compositional Signal Flow Theory", Advancing Research in Information and Communication Technology. IFIP Advances in Information and Communication Technology, IFIP Advances in Information and Communication Technology (Springer) 600: pp. 29–56, doi:10.1007/978-3-030-81701-5_2, ISBN 978-3-030-81700-8, https://inria.hal.science/hal-03325995
14. ↑ Master, Jade; Baez, John C. (2018-08-16). "Open Petri Nets". arXiv:1808.05415v4 [math.CT].
15. ↑ Baez, John C.; Pollard, Blake S. (2018). "A compositional framework for reaction networks". Reviews in Mathematical Physics 29 (9): 1750028–425. doi:10.1142/S0129055X17500283. ISSN 0129-055X. Bibcode: 2017RvMaP..2950028B.
16. ↑ Zardini, Gioele. Co-Design of Complex Systems: From Autonomy to Future Mobility Systems (Thesis). doi:10.3929/ethz-b-000648075.
17. ↑ "The n-Category Café". https://golem.ph.utexas.edu/category/2018/07/symposium_on_compositional_str.html.

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Category theory

Key concepts

| Key concepts|

* Category
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|

Universal constructions|

| Limits|

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Colimits|

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n-categories|

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Strict n-categories|

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