Applied category theory

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] 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 devlope new algorithms about the field.

## List of applied category theorists

[edit]

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

## See also

[edit]

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

## External links

[edit]

Journals:

* Compositionality

Conferences:

* Applied category theory
* Symposium on Compositional Structures (SYCO)[16]

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

[edit]

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. S2CID 199139551.
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. ^ Heunen, Chris; Vicary, Jamie (2019-11-19). Categories for Quantum Theory: An Introduction. ISBN 9780198739616.
10. ^ Coecke, Bob; Sadrzadeh, Mehrnoosh; Clark, Stephen (2011), Mathematical Foundations for a Compositional Distributional Model of Meaning, arXiv:1003.4394
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, arXiv:1401.5980, doi:10.1017/cbo9781139519687.011, ISBN 9781139519687, S2CID 8630039
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, arXiv:1101.0309, doi:10.1007/978-94-007-7284-7_5, ISBN 9789400772830, S2CID 2411818
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, Springer, doi:10.1007/978-3-030-81701-5_2
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\. arXiv:1704.02051. Bibcode:2017RvMaP..2950028B. doi:10.1142/S0129055X17500283. ISSN 0129-055X. S2CID 119665423.
16. ^ "The n-Category Café". golem.ph.utexas.edu. Retrieved 2019-07-20.

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Category theory

Key concepts

| Key concepts|

* Category
* Abelian
* Additive
* Concrete
* Pre-abelian
* Preadditive
* Bicategory
* Adjoint functors
* CCC
* Commutative diagram
* End
* Exponential
* Functor
* Kan extension
* Morphism
* Natural transformation
* Universal property
* Yoneda lemma

|

Universal constructions|

| Limits|

* Terminal objects
* Products
* Equalizers
* Kernels
* Pullbacks
* Inverse limit

|

Colimits|

* 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

Higher category theory

| Key concepts|

* Categorification
* Enriched category
* Higher-dimensional algebra
* Homotopy hypothesis
* Model category
* Simplex category
* String diagram
* Topos

|

n-categories|

| Weak n-categories|

* Bicategory (pseudofunctor)
* Tricategory
* Tetracategory
* Kan complex
* ∞-groupoid
* ∞-topos

|

Strict n-categories|

* 2-category (2-functor)
* 3-category

Categorified concepts|

* 2-group
* 2-ring
* En-ring
* (Traced)(Symmetric) monoidal category
* n-group
* n-monoid

* Category
* Outline
* Glossary

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