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Jonh C, B. (2023). Hoàng Xuân Sính’s thesis: Categorifying Group Theory. Tạp chí Khoa học Thăng Long: Toán và các Khoa học liên quan, 2(8). https://science.thanglong.edu.vn/index.php/volc/article/view/112
Hoàng Xuân Sính’s thesis: Categorifying Group Theory
Nội dung chính của bài viết
Tóm tắt
During what Vietnamese call the American War, Alexander
Grothendieck spent three weeks teaching mathematics in and near Hanoi.
Hoàng Xuân Sính took notes on his lectures and later did her thesis work
with him by correspondence. In her thesis she developed the theory of
‘Gr-categories’, which are monoidal categories in which all objects and
morphisms have inverses. Now often called ‘2-groups’, these structures
allow the study of symmetries that themselves have symmetries. After a
brief account of how Hoàng Xuân Sính wrote her thesis, we explain some
of its main results, and its context in the history of mathematics.
Chi tiết bài viết
Từ khóa
: 2-group, Gr-category, Picard category, monoidal category, symmetric monoidal category, crossed module, 2-group, Gr-category, Picard category, monoidal category, symmetric monoidal category, crossed module
Tài liệu tham khảo
[1] John C. Baez and Aaron D. Lauda, Higher-dimensional algebra V: 2-groups, Theor. Appl. Categ. 12 (2004), 423–491. Also available as arXiv:math.QA/0307200.
[2] John C. Baez and John Huerta, An invitation to higher gauge theory, Gen. Rel. Grav. 43 (2011), 2335–2392. Also available as arXiv:1003.4485.
[3] John C. Baez and Urs Schreiber, Higher gauge theory, in Categories in Algebra, Geometry and Mathematical Physics, eds. Alexei Davydov, Michael Batanin, Michael Johnson, Stephen Lack and Amnon Neeman,Contemp. Math. 431, AMS, Providence, 2007, pp. 7–30. Also available asarXiv:math/0511710.
[4] Maissam Barkeshli, Yu-An Chen, Po-Shen Hsin and Ryohei Kobayashi, Higher-group symmetry in finite gauge theory and stabilizer codes, 2022. Available as arXiv:2211.11764.
[5] Jean Bénabou, Catégories avec multiplication, Comptes Rendus des Séances de l’Académie des Sciences 256 (1963), 1887–1890. Also available as https://gallica.bnf.fr/ark:/12148/bpt6k3208j/f1965.item.
[6] Jean Bénabou, Introduction to bicategories, in Reports of the Midwest Category Seminar, Springer, Berlin, 1967, pp. 1–77.
[7] Arkadiusz Bochniak, Leszek Hadasz, Piotr Korcyl and B la ˙zej Ruba, Study of a lattice 2-group gauge model, in Proceedings of the 38th International Symposium on Lattice Field Theory, LATTICE2021, Proceedings of Science, 2022. Also available as arXiv:2109.12097.
[8] Ronald Brown and C. B. Spencer, G-groupoids, crossed modules, and the classifying space of a topological group, Proc. Kon. Akad. v. Wet. 79 (1976), 296–302. Also available at https://core.ac.uk/download/pdf/82096733.pdf.
[9] Manuel Bullejos and Antonio M. Cegarra, On the geometry of 2-categories and their classifying spaces, K-Theory 29 (2003), 211-229. Also
available at http://www.ugr.es/%7Ebullejos/geometryampl.pdf. 32 John C. Baez
[10] Jack Duskin, Simplicial matrices and the nerves of weak n-categories I: nerves of bicategories, Theory and Applications of Categories, 9 (2001), 198–308. Available at http://www.tac.mta.ca/tac/volumes/9/n10/9-
10abs.html.
[11] Benno Eckmann and Peter H. Hilton, Group-like structures in general categories. I. Multiplications and comultiplications, Math. Ann. 145(3) (1962), 227–255.
[12] Samuel Eilenberg and Saunders Mac Lane, Relations between homology and homotopy groups of spaces,Ann. Math. 46 (2) (1945), 480–509.
[13] Samuel Eilenberg and Saunders Mac Lane, On the groups H(Π, n), I, Ann. Math. 58 (1) (1953), 55–106.
[14] Samuel Eilenberg and Saunders Mac Lane, On the Groups H(Π, n), II: Methods of computation, Ann. Math. 50 (1) (1954), 49–139.
[15] Samuel Eilenberg and G. Max Kelly, Closed categories, Proceedings of the Conference on Categorical Algebra: La Jolla 1965, Springer, 1966.
[16] Magnus Forrester-Barker, Group objects and internal categories. Available as math.CT/0212065.
[17] Alexander Grothendieck, Pursuing Stacks, 1984. Available as arXiv:2111.01000.
[18] Alexander Grothendieck, Récoltes et Semailles: Réflexions et témoignages sur un passé de mathématicien, Gallimard, Paris, 2022. Also available at https://webusers.imjprg.fr/ leila.schneps/grothendieckcircle/writings.php.
[19] K. A. Hardie, K. H. Kamps and R. W. Kieboom, A homotopy 2-groupoid of a topological space, Appl. Cat. Str. 8 (2000) 209–234.
[20] K. A. Hardie, K. H. Kamps and R. W. Kieboom, A homotopy bigroupoid of a topological space, Appl. Cat. Str. 9 (2001) 311–327.
[21] Simon Henry and Edoardo Lanari, On the homotopy hypothesis in dimension 3. Available as arXiv:1905.05625.
[22] Peter Hilton, The birth of homological algebra, Rocky Mountain J. Math. 32 (4) (2002), 1101–1116. Also available at https://projecteuclid.org/journals/rocky-mountain-journalof-mathematics/volume-32/issue-4/The-Birth-of-HomologicalAlgebra/10.1216/rmjm/1181070011.full. Hoàng Xuân Sính’s Thesis: Categorifying Group Theory 33
[23] Hoàng Xuân Sính, Gr-catégories, Ph.D. thesis, Université Paris VII, 1973. Handwritten version available at https://pnp.mathematik.uni stuttgart.de/lexmath/kuenzer/sinh.html. Version typeset by Cristian David Gonzalez Avilés available at https://agrothendieck.github.io/divers/GCS.pdf.
[24] Hoàng Xuân Sính, Gr-categories: summary. Handwritten version available at https://pnp.mathematik.unistuttgart.de/lexmath/kuenzer/sinh.html.
[25] Hoàng Xuân Sính, Gr-catégories strictes, Acta Math. Vietnam. 3 (2) (1978), 47–59. Also available at https://pnp.mathematik.unistuttgart.de/lexmath/kuenzer/Hoang_Xuan_Sinh_Gr_categories_strictes.pdf.
[26] Hoàng Xuân Sính, Catégories de Picard restreintes, Acta Math. Vietnam. 7 (1) (1983), 117–122. Also available at https://pnp.mathematik.unistuttgart.de/lexmath/kuenzer/Hoang_Xuan_Sinh_Categories_de_Picard_restreintes.pdf.
[27] Hoàng Xuân Sính, personal communication (translated from French), July 7, 2023.
[28] G. Max Kelly, On MacLane’s conditions for coherence of natural associativities, commutativities, etc., J. Alg. 1 (1964), 397–402. Also available at https://www.sciencedirect.com/science/article/pii/0021869364900183.
[29] André Joyal and Ross Street, Braided monoidal categories, Macquarie Mathematics Report No. 860081, November 1986. Available at http://web.science.mq.edu.au ∼street/JS1.pdf.
[30] Saunders Mac Lane, Natural associativity and commutativity, Rice University Studies 49 (4) (1963), 28–46. Available as https://scholarship.rice.edu/handle/1911/62865.
[31] Saunders Mac Lane, Origins of the cohomology of groups, Enseign. Math. 24 (1978), 191–219.
[32] Saunders Mac Lane, Historical note, Jour. Alg. 60 (2) (1979), 319–320.Also available athttps://www.sciencedirect.com/science/article/pii/0021869379900851.
[33] Saunders Mac Lane, Group extensions for 45 years, Math. Intell. 10(2) (1988), 29–35.
[34] Saunders Mac Lane, Categories for the Working Mathematician, Springer, Berlin, 2013. 34 John C. Baez
[35] Saunders Mac Lane and J. H. C. Whitehead, On the 3-type of a complex, Proc. Nat. Acad. Sci. 36 (1950), 41–48. Also available at https://www.pnas.org/doi/pdf/10.1073/pnas.36.1.41.
[36] Fernando Muro and Andrew Tonks, Unital associahedra, Forum Math. 26 (2) (2014), 593–620. Also available as arXiv:1110.1959.
[37] Thomas Nikolaus, Urs Schreiber and Danny Stevenson, Principal ∞-bundles: general theory, J. Homotopy Relat. Struct. 10 (4) (2015), 749–801. Also available as arXiv:1207.0248.
[38] Behrang Noohi, Notes on 2-groupoids, 2-groups and crossed-modules, Homology Homotopy Appl. 9 (1) 2008, 75–106. Also available as arXiv:math/0512106.
[39] Joseph J. Rotman, Introduction to Homological Algebra, Academic Press, New York, 1979.
[40] Neantro Saavedra-Rivano, Catégories Tannakienes, Ph.D. thesis, Université Paris VII, 1970.
[41] Neantro Saavedra-Rivano, Catégories Tannakienes, Springer Lecture Notes in Mathematics 265, Springer, Berlin, 1972. Related version available at https://eudml.org/doc/87193.
[42] Urs Schreiber, Differential cohomology in a cohesive infinity-topos, 2013. Available as arXiv:1310.7930.
[43] Alexandru Solian, Coherence in categorical groups, Comm. Alg. 9 (1981), 1039–1057.
[44] James Stasheff, Homotopy associative H-spaces I, II, Trans. Amer. Math. Soc. 108 (1963), 275–312.
[45] An Thanh, Hoàng Xuân Sính với "Ý tưởng lãng mạn nhất cuộc đời", Thông Tin Đối Ngoại, November 21, 2019. Available at https://web.archive.org/web/20190926174449/http:/tapchithongtindoingoai.vn/kieu-bao-huong-ve-to-quoc/gs-hoang-xuansinh-voi-y-tuong-lang-man-nhat-cuoc-doi-19467.
[46] Todd Trimble, Combinatorics of polyhedra forn-categories, September 5, 1999. Available as https://math.ucr.edu/home/baez/trimble/polyhedra.html.
[47] Charles A. Weibel, History of homological algebra, in The History of Topology, ed. Ioan M. James, Elsevier, 1999. Also available as https://conf.math.illinois.edu/K theory/0245/. Hoàng Xuân Sính’s Thesis: Categorifying Group Theory 35
[48] J. H. C. Whitehead, Note on a previous paper entitled ‘On adding relations to homotopy groups’, Ann. Math. 47 (1946), 806–810.
[49] J. H. C. Whitehead, Combinatorial homotopy II, Bull. Amer. Math. Soc. 55 (1949), 453–496. Also available at https://projecteuclid.org/journals/bulletin-of-theamerican-mathematical-society/volume-55/issue-5/Combinatorialhomotopy-II/bams/1183513797.full. John C. Baez
[2] John C. Baez and John Huerta, An invitation to higher gauge theory, Gen. Rel. Grav. 43 (2011), 2335–2392. Also available as arXiv:1003.4485.
[3] John C. Baez and Urs Schreiber, Higher gauge theory, in Categories in Algebra, Geometry and Mathematical Physics, eds. Alexei Davydov, Michael Batanin, Michael Johnson, Stephen Lack and Amnon Neeman,Contemp. Math. 431, AMS, Providence, 2007, pp. 7–30. Also available asarXiv:math/0511710.
[4] Maissam Barkeshli, Yu-An Chen, Po-Shen Hsin and Ryohei Kobayashi, Higher-group symmetry in finite gauge theory and stabilizer codes, 2022. Available as arXiv:2211.11764.
[5] Jean Bénabou, Catégories avec multiplication, Comptes Rendus des Séances de l’Académie des Sciences 256 (1963), 1887–1890. Also available as https://gallica.bnf.fr/ark:/12148/bpt6k3208j/f1965.item.
[6] Jean Bénabou, Introduction to bicategories, in Reports of the Midwest Category Seminar, Springer, Berlin, 1967, pp. 1–77.
[7] Arkadiusz Bochniak, Leszek Hadasz, Piotr Korcyl and B la ˙zej Ruba, Study of a lattice 2-group gauge model, in Proceedings of the 38th International Symposium on Lattice Field Theory, LATTICE2021, Proceedings of Science, 2022. Also available as arXiv:2109.12097.
[8] Ronald Brown and C. B. Spencer, G-groupoids, crossed modules, and the classifying space of a topological group, Proc. Kon. Akad. v. Wet. 79 (1976), 296–302. Also available at https://core.ac.uk/download/pdf/82096733.pdf.
[9] Manuel Bullejos and Antonio M. Cegarra, On the geometry of 2-categories and their classifying spaces, K-Theory 29 (2003), 211-229. Also
available at http://www.ugr.es/%7Ebullejos/geometryampl.pdf. 32 John C. Baez
[10] Jack Duskin, Simplicial matrices and the nerves of weak n-categories I: nerves of bicategories, Theory and Applications of Categories, 9 (2001), 198–308. Available at http://www.tac.mta.ca/tac/volumes/9/n10/9-
10abs.html.
[11] Benno Eckmann and Peter H. Hilton, Group-like structures in general categories. I. Multiplications and comultiplications, Math. Ann. 145(3) (1962), 227–255.
[12] Samuel Eilenberg and Saunders Mac Lane, Relations between homology and homotopy groups of spaces,Ann. Math. 46 (2) (1945), 480–509.
[13] Samuel Eilenberg and Saunders Mac Lane, On the groups H(Π, n), I, Ann. Math. 58 (1) (1953), 55–106.
[14] Samuel Eilenberg and Saunders Mac Lane, On the Groups H(Π, n), II: Methods of computation, Ann. Math. 50 (1) (1954), 49–139.
[15] Samuel Eilenberg and G. Max Kelly, Closed categories, Proceedings of the Conference on Categorical Algebra: La Jolla 1965, Springer, 1966.
[16] Magnus Forrester-Barker, Group objects and internal categories. Available as math.CT/0212065.
[17] Alexander Grothendieck, Pursuing Stacks, 1984. Available as arXiv:2111.01000.
[18] Alexander Grothendieck, Récoltes et Semailles: Réflexions et témoignages sur un passé de mathématicien, Gallimard, Paris, 2022. Also available at https://webusers.imjprg.fr/ leila.schneps/grothendieckcircle/writings.php.
[19] K. A. Hardie, K. H. Kamps and R. W. Kieboom, A homotopy 2-groupoid of a topological space, Appl. Cat. Str. 8 (2000) 209–234.
[20] K. A. Hardie, K. H. Kamps and R. W. Kieboom, A homotopy bigroupoid of a topological space, Appl. Cat. Str. 9 (2001) 311–327.
[21] Simon Henry and Edoardo Lanari, On the homotopy hypothesis in dimension 3. Available as arXiv:1905.05625.
[22] Peter Hilton, The birth of homological algebra, Rocky Mountain J. Math. 32 (4) (2002), 1101–1116. Also available at https://projecteuclid.org/journals/rocky-mountain-journalof-mathematics/volume-32/issue-4/The-Birth-of-HomologicalAlgebra/10.1216/rmjm/1181070011.full. Hoàng Xuân Sính’s Thesis: Categorifying Group Theory 33
[23] Hoàng Xuân Sính, Gr-catégories, Ph.D. thesis, Université Paris VII, 1973. Handwritten version available at https://pnp.mathematik.uni stuttgart.de/lexmath/kuenzer/sinh.html. Version typeset by Cristian David Gonzalez Avilés available at https://agrothendieck.github.io/divers/GCS.pdf.
[24] Hoàng Xuân Sính, Gr-categories: summary. Handwritten version available at https://pnp.mathematik.unistuttgart.de/lexmath/kuenzer/sinh.html.
[25] Hoàng Xuân Sính, Gr-catégories strictes, Acta Math. Vietnam. 3 (2) (1978), 47–59. Also available at https://pnp.mathematik.unistuttgart.de/lexmath/kuenzer/Hoang_Xuan_Sinh_Gr_categories_strictes.pdf.
[26] Hoàng Xuân Sính, Catégories de Picard restreintes, Acta Math. Vietnam. 7 (1) (1983), 117–122. Also available at https://pnp.mathematik.unistuttgart.de/lexmath/kuenzer/Hoang_Xuan_Sinh_Categories_de_Picard_restreintes.pdf.
[27] Hoàng Xuân Sính, personal communication (translated from French), July 7, 2023.
[28] G. Max Kelly, On MacLane’s conditions for coherence of natural associativities, commutativities, etc., J. Alg. 1 (1964), 397–402. Also available at https://www.sciencedirect.com/science/article/pii/0021869364900183.
[29] André Joyal and Ross Street, Braided monoidal categories, Macquarie Mathematics Report No. 860081, November 1986. Available at http://web.science.mq.edu.au ∼street/JS1.pdf.
[30] Saunders Mac Lane, Natural associativity and commutativity, Rice University Studies 49 (4) (1963), 28–46. Available as https://scholarship.rice.edu/handle/1911/62865.
[31] Saunders Mac Lane, Origins of the cohomology of groups, Enseign. Math. 24 (1978), 191–219.
[32] Saunders Mac Lane, Historical note, Jour. Alg. 60 (2) (1979), 319–320.Also available athttps://www.sciencedirect.com/science/article/pii/0021869379900851.
[33] Saunders Mac Lane, Group extensions for 45 years, Math. Intell. 10(2) (1988), 29–35.
[34] Saunders Mac Lane, Categories for the Working Mathematician, Springer, Berlin, 2013. 34 John C. Baez
[35] Saunders Mac Lane and J. H. C. Whitehead, On the 3-type of a complex, Proc. Nat. Acad. Sci. 36 (1950), 41–48. Also available at https://www.pnas.org/doi/pdf/10.1073/pnas.36.1.41.
[36] Fernando Muro and Andrew Tonks, Unital associahedra, Forum Math. 26 (2) (2014), 593–620. Also available as arXiv:1110.1959.
[37] Thomas Nikolaus, Urs Schreiber and Danny Stevenson, Principal ∞-bundles: general theory, J. Homotopy Relat. Struct. 10 (4) (2015), 749–801. Also available as arXiv:1207.0248.
[38] Behrang Noohi, Notes on 2-groupoids, 2-groups and crossed-modules, Homology Homotopy Appl. 9 (1) 2008, 75–106. Also available as arXiv:math/0512106.
[39] Joseph J. Rotman, Introduction to Homological Algebra, Academic Press, New York, 1979.
[40] Neantro Saavedra-Rivano, Catégories Tannakienes, Ph.D. thesis, Université Paris VII, 1970.
[41] Neantro Saavedra-Rivano, Catégories Tannakienes, Springer Lecture Notes in Mathematics 265, Springer, Berlin, 1972. Related version available at https://eudml.org/doc/87193.
[42] Urs Schreiber, Differential cohomology in a cohesive infinity-topos, 2013. Available as arXiv:1310.7930.
[43] Alexandru Solian, Coherence in categorical groups, Comm. Alg. 9 (1981), 1039–1057.
[44] James Stasheff, Homotopy associative H-spaces I, II, Trans. Amer. Math. Soc. 108 (1963), 275–312.
[45] An Thanh, Hoàng Xuân Sính với "Ý tưởng lãng mạn nhất cuộc đời", Thông Tin Đối Ngoại, November 21, 2019. Available at https://web.archive.org/web/20190926174449/http:/tapchithongtindoingoai.vn/kieu-bao-huong-ve-to-quoc/gs-hoang-xuansinh-voi-y-tuong-lang-man-nhat-cuoc-doi-19467.
[46] Todd Trimble, Combinatorics of polyhedra forn-categories, September 5, 1999. Available as https://math.ucr.edu/home/baez/trimble/polyhedra.html.
[47] Charles A. Weibel, History of homological algebra, in The History of Topology, ed. Ioan M. James, Elsevier, 1999. Also available as https://conf.math.illinois.edu/K theory/0245/. Hoàng Xuân Sính’s Thesis: Categorifying Group Theory 35
[48] J. H. C. Whitehead, Note on a previous paper entitled ‘On adding relations to homotopy groups’, Ann. Math. 47 (1946), 806–810.
[49] J. H. C. Whitehead, Combinatorial homotopy II, Bull. Amer. Math. Soc. 55 (1949), 453–496. Also available at https://projecteuclid.org/journals/bulletin-of-theamerican-mathematical-society/volume-55/issue-5/Combinatorialhomotopy-II/bams/1183513797.full. John C. Baez