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{{Cite web|url=https://ocw.mit.edu/courses/mathematics/18-s097-applied-category-theory-january-iap-2019/|title=Applied Category Theory|website=MIT OpenCourseWare|access-date=2019-07-20}}{{Cite book|title=An Invitation to Applied Category Theory by Brendan Fong|last1=Spivak|first1=David I.|last2=Fong|first2=Brendan|date=July 2019|doi=10.1017/9781108668804|isbn=9781108668804|s2cid=199139551}}{{Cite arXiv|last=Bradley|first=Tai-Danae|date=2018-09-16|title=What is Applied Category Theory?|class=math.CT|eprint=1809.05923v2}} including but not limited to computer science,{{Cite book|title=Category theory for computing science|last=Barr, Michael.|authorlink = Michael Barr (mathematician)|date=1990|publisher=Prentice Hall|others=Wells, Charles.|isbn=0131204866|location=New York|oclc=19126000}}{{Cite journal|last1=Ehrig|first1=Hartmut| author-link=Hartmut Ehrig|last2=Große-Rhode|first2=Martin|last3=Wolter|first3=Uwe|date=1998-03-01|title=Applications of Category Theory to the Area of Algebraic Specification in Computer Science|journal=Applied Categorical Structures|volume=6|issue=1|pages=1–35|doi=10.1023/A:1008688122154|s2cid=290074|issn=1572-9095}} physics (in particular quantum mechanics{{Citation|last1=Abramsky|first1=Samson|authorlink = Samson Abramsky|title=Categorical Quantum Mechanics|date=2009|work=Handbook of Quantum Logic and Quantum Structures|pages=261–323|publisher=Elsevier|isbn=9780444528698|last2=Coecke|first2=Bob|author2link = Bob Coecke|doi=10.1016/b978-0-444-52869-8.50010-4|arxiv=0808.1023|s2cid=692816}}{{Cite journal|last1=Duncan|first1=Ross|last2=Coecke|first2=Bob|author2link = Bob Coecke|year=2011|title=Interacting Quantum Observables: Categorical Algebra and Diagrammatics|journal=New Journal of Physics|volume=13|issue=4|pages=043016|doi=10.1088/1367-2630/13/4/043016|arxiv=0906.4725|bibcode=2011NJPh...13d3016C|s2cid=14259278}}{{Cite book|title=Picturing quantum processes : a first course in quantum theory and diagrammatic reasoning|last1=Coecke|first1=Bob|author1link = Bob Coecke|last2=Kissinger|first2=Aleks|isbn=978-1107104228|oclc=1026174191|date = 2017-03-16}}{{Cite book|title=Categories for Quantum Theory: An Introduction|last1=Heunen|first1=Chris|last2=Vicary|first2=Jamie|isbn=9780198739616|date = 2019-11-19}}), natural language processing,{{Citation|last1=Coecke|first1=Bob|author1link = Bob Coecke|year=2011|title=Mathematical Foundations for a Compositional Distributional Model of Meaning|arxiv=1003.4394|last2=Sadrzadeh|first2=Mehrnoosh|last3=Clark|first3=Stephen}}{{Citation|last1=Kartsaklis|first1=Dimitri|chapter=Reasoning about meaning in natural language with compact closed categories and Frobenius algebras|pages=199–222|publisher=Cambridge University Press|isbn=9781139519687|last2=Sadrzadeh|first2=Mehrnoosh|last3=Pulman|first3=Stephen|last4=Coecke|first4=Bob|author4link = Bob Coecke|doi=10.1017/cbo9781139519687.011|title=Logic and Algebraic Structures in Quantum Computing|year=2016|arxiv=1401.5980|s2cid=8630039}}{{Citation|last1=Grefenstette|first1=Edward|title=Concrete Sentence Spaces for Compositional Distributional Models of Meaning|date=2014|work=Text, Speech and Language Technology|pages=71–86|publisher=Springer Netherlands|isbn=9789400772830|last2=Sadrzadeh|first2=Mehrnoosh|last3=Clark|first3=Stephen|last4=Coecke|first4=Bob|author4link = Bob Coecke|last5=Pulman|first5=Stephen|doi=10.1007/978-94-007-7284-7_5|arxiv=1101.0309|s2cid=2411818}} control theory,{{Citation|last1=Bonchi|first1=Filippo|last2=Sobocinski|first2=Pawel|last3=Zanasi|first3=Fabio|year=2021|work= Advancing Research in Information and Communication Technology. IFIP Advances in Information and Communication Technology |title=A Survey of Compositional Signal Flow Theory|publisher=Springer|doi=10.1007/978-3-030-81701-5_2}}{{Cite arXiv|last1=Master|first1=Jade|last2=Baez|first2=John C.|author2link = John Baez|date=2018-08-16|title=Open Petri Nets|class=math.CT|eprint=1808.05415v4}}{{Cite journal|last1=Baez|first1=John C.|author1link = John Baez|last2=Pollard|first2=Blake S.|year=2018|title=A compositional framework for reaction networks|journal=Reviews in Mathematical Physics|volume=29|issue=9|pages=1750028–425|doi=10.1142/S0129055X17500283|issn=0129-055X|bibcode=2017RvMaP..2950028B|arxiv=1704.02051|s2cid=119665423}} 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.