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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]), control theory,[9][10] natural language processing,[11][12] 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.
See also

Categorical quantum mechanics
Petri net
Univalent foundations
String diagrams

External links


Applied categorical structures


Applied category theory
Symposium on Compositional Structures (SYCO)[13]


An Invitation to Applied Category Theory


"Applied Category Theory". MIT OpenCourseWare. Retrieved 2019-07-20.
Spivak, David I.; Fong, Brendan (July 2019). An Invitation to Applied Category Theory by Brendan Fong. doi:10.1017/9781108668804. ISBN 9781108668804.
Bradley, Tai-Danae (2018-09-16). "What is Applied Category Theory?". arXiv:1809.05923v2 [math.CT].
Barr, Michael. (1990). Category theory for computing science. Wells, Charles. New York: Prentice Hall. ISBN 0131204866. OCLC 19126000.
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. S2CID 290074.
Abramsky, Samson; Coecke, Bob (2009), "Categorical Quantum Mechanics", Handbook of Quantum Logic and Quantum Structures, Elsevier, pp. 261–323, arXiv:0808.1023, doi:10.1016/b978-0-444-52869-8.50010-4, ISBN 9780444528698, S2CID 692816
Duncan, Ross; Coecke, Bob (2011). "Interacting Quantum Observables: Categorical Algebra and Diagrammatics". New Journal of Physics. 13 (4): 043016. arXiv:0906.4725. Bibcode:2011NJPh...13d3016C. doi:10.1088/1367-2630/13/4/043016. S2CID 14259278.
Coecke, Bob. (2017-03-16). Picturing quantum processes : a first course in quantum theory and diagrammatic reasoning. ISBN 978-1107104228. OCLC 1026174191.
Master, Jade; Baez, John C. (2018-08-16). "Open Petri Nets". arXiv:1808.05415v4 [math.CT].
Baez, John C.; Pollard, Blake S. (2018). "A compositional framework for reaction networks". Reviews in Mathematical Physics. 29 (9): 1750028–425. arXiv:1704.02051. Bibcode:2017RvMaP..2950028B. doi:10.1142/S0129055X17500283. ISSN 0129-055X. S2CID 119665423.
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, arXiv:1401.5980, doi:10.1017/cbo9781139519687.011, ISBN 9781139519687, S2CID 8630039
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, arXiv:1101.0309, doi:10.1007/978-94-007-7284-7_5, ISBN 9789400772830, S2CID 2411818

"The n-Category Café". golem.ph.utexas.edu. Retrieved 2019-07-20.

Category theory
Key concepts
Key concepts

Category Adjoint functors CCC Commutative diagram Concrete category End Exponential Functor Kan extension Morphism Natural transformation Universal property

Universal constructions

Terminal objects Products Equalizers
Kernels Pullbacks Inverse limit


Initial objects Coproducts Coequalizers
Cokernels and quotients Pushout Direct limit

Algebraic categories

Sets Relations Magmas Groups Abelian groups Rings (Fields) Modules (Vector spaces)

Constructions on categories

Free category Functor category Kleisli category Opposite category Quotient category Product category Comma category Subcategory

A simple triangular commutative diagram
Higher category theory


Mathematics Encyclopedia



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