Applied category theory

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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] [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 about the field.

Contents

List of applied category theorists

See also

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<span class="mw-page-title-main">John C. Baez</span> American mathematician

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<span class="mw-page-title-main">Samson Abramsky</span> British computer scientist

Samson Abramsky is Professor of Computer Science at University College London. He was previously the Christopher Strachey Professor of Computing at Wolfson College, Oxford, from 2000 to 2021.

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<span class="mw-page-title-main">André Joyal</span> Canadian mathematician

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Basil J. Hiley, is a British quantum physicist and professor emeritus of the University of London.

<span class="mw-page-title-main">Bob Coecke</span>

Bob Coecke is a Belgian theoretical physicist and logician who was professor of Quantum foundations, Logics and Structures at Oxford University until 2020, when he became Chief Scientist of Cambridge Quantum Computing, and after the merger with Honeywell Quantum Systems, Chief Scientist of Quantinuum. In January 2023 he also became Distinguished Visiting Research Chair at the Perimeter Institute for Theoretical Physics. He pioneered categorical quantum mechanics, Quantum Picturalism, ZX-calculus, DisCoCat model for natural language, and quantum natural language processing (QNLP). He is a founder of the Quantum Physics and Logic community and conference series, and of the applied category theory community, conference series, and diamond-open-access journal Compositionality.

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In mathematics, Solèr's theorem is a result concerning certain infinite-dimensional vector spaces. It states that any orthomodular form that has an infinite orthonormal set is a Hilbert space over the real numbers, complex numbers or quaternions. Originally proved by Maria Pia Solèr, the result is significant for quantum logic and the foundations of quantum mechanics. In particular, Solèr's theorem helps to fill a gap in the effort to use Gleason's theorem to rederive quantum mechanics from information-theoretic postulates. It is also an important step in the Heunen–Kornell axiomatisation of the category of Hilbert spaces.

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The ZX-calculus is a rigorous graphical language for reasoning about linear maps between qubits, which are represented as string diagrams called ZX-diagrams. A ZX-diagram consists of a set of generators called spiders that represent specific tensors. These are connected together to form a tensor network similar to Penrose graphical notation. Due to the symmetries of the spiders and the properties of the underlying category, topologically deforming a ZX-diagram does not affect the linear map it represents. In addition to the equalities between ZX-diagrams that are generated by topological deformations, the calculus also has a set of graphical rewrite rules for transforming diagrams into one another. The ZX-calculus is universal in the sense that any linear map between qubits can be represented as a diagram, and different sets of graphical rewrite rules are complete for different families of linear maps. ZX-diagrams can be seen as a generalisation of quantum circuit notation.

<span class="mw-page-title-main">David Spivak</span> American mathematician

David Isaac Spivak is an American mathematician and senior scientist at the Topos Institute. He has worked on applications of category theory, in particular ologs and operadic compositionality of dynamical systems. He authored and coauthored the introductory texts on category theory and its applications, Category Theory for the Sciences and An Invitation to Applied Category Theory.

In mathematics, nonabelian algebraic topology studies an aspect of algebraic topology that involves higher-dimensional algebras.

DisCoCat is a mathematical framework for natural language processing which uses category theory to unify distributional semantics with the principle of compositionality. The grammatical derivations in a categorial grammar are interpreted as linear maps acting on the tensor product of word vectors to produce the meaning of a sentence or a piece of text. String diagrams are used to visualise information flow and reason about natural language semantics.

Quantum natural language processing (QNLP) is the application of quantum computing to natural language processing (NLP). It computes word embeddings as parameterised quantum circuits that can solve NLP tasks faster than any classical computer. It is inspired by categorical quantum mechanics and the DisCoCat framework, making use of string diagrams to translate from grammatical structure to quantum processes.

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. 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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