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'''Applied category theory''' is an [[Discipline (academia)|academic discipline]] in which methods from [[category theory]] are used to study other fields<ref>{{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}}</ref><ref>{{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}}</ref><ref>{{Cite arxiv|last=Bradley|first=Tai-Danae|date=2018-09-16|title=What is Applied Category Theory?|class=math.CT|eprint=1809.05923v2}}</ref> including but not limited to [[computer science]],<ref>{{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}}</ref><ref>{{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}}</ref> [[physics]] (in particular [[Categorical quantum mechanics|quantum mechanics]]<ref>{{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}}</ref><ref>{{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}}</ref><ref>{{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}}</ref>), [[natural language processing]], <ref>{{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}}</ref><ref>{{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}}</ref><ref>{{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}}</ref>, [[control theory]]<ref>{{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}}</ref><ref>{{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}}</ref>, [[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.
'''Applied category theory''' is an [[Discipline (academia)|academic discipline]] in which methods from [[category theory]] are used to study other fields<ref>{{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}}</ref><ref>{{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}}</ref><ref>{{Cite arxiv|last=Bradley|first=Tai-Danae|date=2018-09-16|title=What is Applied Category Theory?|class=math.CT|eprint=1809.05923v2}}</ref> including but not limited to [[computer science]],<ref>{{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}}</ref><ref>{{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}}</ref> [[physics]] (in particular [[Categorical quantum mechanics|quantum mechanics]]<ref>{{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}}</ref><ref>{{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}}</ref><ref>{{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}}</ref>), [[natural language processing]],<ref>{{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}}</ref><ref>{{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}}</ref><ref>{{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}}</ref> [[control theory]],<ref>{{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}}</ref><ref>{{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}}</ref> [[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 ==
== See also ==

Revision as of 12:44, 30 August 2021

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,[9][10][11] control theory,[12][13] 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

Journals:

Conferences:

Books:

Institutes:

References

  1. ^ "Applied Category Theory". MIT OpenCourseWare. Retrieved 2019-07-20.
  2. ^ Spivak, David I.; Fong, Brendan (July 2019). An Invitation to Applied Category Theory by Brendan Fong. doi:10.1017/9781108668804. ISBN 9781108668804.
  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 0131204866. 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. S2CID 290074.
  6. ^ 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
  7. ^ 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.
  8. ^ Coecke, Bob; Kissinger, Aleks (2017-03-16). Picturing quantum processes : a first course in quantum theory and diagrammatic reasoning. ISBN 978-1107104228. OCLC 1026174191.
  9. ^ Coecke, Bob; Sadrzadeh, Mehrnoosh; Clark, Stephen (2011), Mathematical Foundations for a Compositional Distributional Model of Meaning, arXiv:1003.4394
  10. ^ 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
  11. ^ 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
  12. ^ Master, Jade; Baez, John C. (2018-08-16). "Open Petri Nets". arXiv:1808.05415v4 [math.CT].
  13. ^ 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.
  14. ^ "The n-Category Café". golem.ph.utexas.edu. Retrieved 2019-07-20.
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