Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Model theory and the Tannakian formalism


Author: Moshe Kamensky
Journal: Trans. Amer. Math. Soc. 367 (2015), 1095-1120
MSC (2010): Primary 03C40, 14L17, 18D10, 20G05
DOI: https://doi.org/10.1090/S0002-9947-2014-06062-5
Published electronically: October 10, 2014
MathSciNet review: 3280038
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We draw the connection between the model theoretic notions of internality and the binding group on one hand, and the Tannakian formalism on the other.

More precisely, we deduce the fundamental results of the Tannakian formalism by associating to a Tannakian category a first order theory, and applying the results on internality there. We then formulate the notion of a differential tensor category, which axiomatises the category of differential representations of differential linear groups, and show how the model theoretic techniques can be used to deduce the analogous results in that context.


References [Enhancements On Off] (What's this?)

  • [1] Phyllis Joan Cassidy, Differential algebraic groups, Amer. J. Math. 94 (1972), 891-954. MR 0360611 (50:13058)
  • [2] Phyllis Joan Cassidy, The differential rational representation algebra on a linear differential algebraic group, J. Algebra 37 (1975), no. 2, 223-238. MR 0409426 (53 #13181)
  • [3] P. Deligne, Catégories tannakiennes, The Grothendieck Festschrift, Vol. II, Progr. Math., vol. 87, Birkhäuser Boston, Boston, MA, 1990, pp. 111-195 (French). MR 1106898 (92d:14002)
  • [4] Pierre Deligne, James S. Milne, Arthur Ogus, and Kuang-yen Shih, Hodge cycles, motives, and Shimura varieties, Lecture Notes in Mathematics, vol. 900, Springer-Verlag, Berlin, 1982. MR 654325 (84m:14046)
  • [5] Peter Freyd, Abelian categories. An introduction to the theory of functors, Harper's Series in Modern Mathematics, Harper & Row Publishers, New York, 1964. MR 0166240 (29 #3517)
  • [6] Alexander Grothendieck et al., Revêtements étales et groupe fondamental (SGA 1), Documents Mathématiques (Paris) [Mathematical Documents (Paris)], 3, Société Mathématique de France, Paris, 2003 (French). Séminaire de géométrie algébrique du Bois Marie 1960-61. [Algebraic Geometry Seminar of Bois Marie 1960-61]; Directed by A. Grothendieck; With two papers by M. Raynaud; Updated and annotated reprint of the 1971 original [Lecture Notes in Math., 224, Springer, Berlin; MR0354651 (50 #7129)]. MR 2017446 (2004g:14017)
  • [7] Bradd Hart and Ziv Shami, On the type-definability of the binding group in simple theories, J. Symbolic Logic 70 (2005), no. 2, 379-388. MR 2140036 (2006b:03037), https://doi.org/10.2178/jsl/1120224718
  • [8] Ehud Hrushovski, Computing the Galois group of a linear differential equation, Differential Galois theory (Bedlewo, 2001) Banach Center Publ., vol. 58, Polish Acad. Sci., Warsaw, 2002, pp. 97-138. MR 1972449 (2004j:12007), https://doi.org/10.4064/bc58-0-9
  • [9] Ehud Hrushovski. Groupoids, imaginaries and internal covers, Turkish J. Math. 36 (2012), no. 2, 173-198. MR 2912035
  • [10] Ehud Hrushovski, Unidimensional theories are superstable, Ann. Pure Appl. Logic 50 (1990), no. 2, 117-138. MR 1081816 (92g:03052), https://doi.org/10.1016/0168-0072(90)90046-5
  • [11] Moshe Kamensky, Models, logics, and higher-dimensional categories, CRM Proceedings & Lecture Notes, vol. 53, American Mathematical Society, Providence, RI, 2011. A tribute to the work of Mihály Makkai; Edited by Bradd Hart, Thomas G. Kucera, Anand Pillay, Philip J. Scott and Robert A. G. Seely. MR 2848606 (2012i:00026)
  • [12] Moshe Kamensky, Tannakian formalism over fields with operators, Int. Math. Res. Not. IMRN 2013, no. 24, 5571-5622. MR 3144174
  • [13] E. R. Kolchin, Differential algebra and algebraic groups, Academic Press, New York, 1973. Pure and Applied Mathematics, Vol. 54. MR 0568864 (58 #27929)
  • [14] Saunders Mac Lane, Categories for the working mathematician, 2nd ed., Graduate Texts in Mathematics, vol. 5, Springer-Verlag, New York, 1998. MR 1712872 (2001j:18001)
  • [15] David Marker, Model theory, Graduate Texts in Mathematics, vol. 217, Springer-Verlag, New York, 2002. An introduction. MR 1924282 (2003e:03060)
  • [16] David Marker, Model theory of differential fields, Model theory, algebra, and geometry, Math. Sci. Res. Inst. Publ., vol. 39, Cambridge Univ. Press, Cambridge, 2000, pp. 53-63. MR 1773702 (2001j:12005)
  • [17] David Marker, Margit Messmer, and Anand Pillay, Model theory of fields, 2nd ed., Lecture Notes in Logic, vol. 5, Association for Symbolic Logic, La Jolla, CA, 2006. MR 2215060 (2006k:03063)
  • [18] James S. Milne, Étale cohomology, Princeton Mathematical Series, vol. 33, Princeton University Press, Princeton, N.J., 1980. MR 559531 (81j:14002)
  • [19] Alexey Ovchinnikov, Tannakian approach to linear differential algebraic groups, Transform. Groups 13 (2008), no. 2, 413-446. MR 2426137 (2010h:20113), https://doi.org/10.1007/s00031-008-9010-4
  • [20] Alexey Ovchinnikov, Tannakian categories, linear differential algebraic groups, and parametrized linear differential equations, Transform. Groups 14 (2009), no. 1, 195-223. MR 2480859 (2010b:18012), https://doi.org/10.1007/s00031-008-9042-9
  • [21] Anand Pillay, Algebraic $ D$-groups and differential Galois theory, Pacific J. Math. 216 (2004), no. 2, 343-360. MR 2094550 (2005k:12007)
  • [22] A. Pillay, Model theory, Course lecture notes. 2002. URL: http://www.math.uiuc. $ \linebreak$edu/People/pillay/lecturenotes_modeltheory.pdf (cit. on pp. 3, 4).
  • [23] Anand Pillay, Some foundational questions concerning differential algebraic groups, Pacific J. Math. 179 (1997), no. 1, 179-200. MR 1452531 (98g:12008), https://doi.org/10.2140/pjm.1997.179.179
  • [24] Bruno Poizat, Stable groups, Mathematical Surveys and Monographs, vol. 87, American Mathematical Society, Providence, RI, 2001. Translated from the 1987 French original by Moses Gabriel Klein. MR 1827833 (2002a:03067)
  • [25] Bruno Poizat, Une théorie de Galois imaginaire, J. Symbolic Logic 48 (1983), no. 4, 1151-1170 (1984) (French). MR 727805 (85e:03083), https://doi.org/10.2307/2273680
  • [26] Neantro Saavedra Rivano, Catégories Tannakiennes, Lecture Notes in Mathematics, Vol. 265, Springer-Verlag, Berlin, 1972 (French). MR 0338002 (49 #2769)
  • [27] William C. Waterhouse, Introduction to affine group schemes, Graduate Texts in Mathematics, vol. 66, Springer-Verlag, New York, 1979. MR 547117 (82e:14003)
  • [28] B. I. Zilber, Totally categorical theories: structural properties and the nonfinite axiomatizability, Model theory of algebra and arithmetic (Proc. Conf., Karpacz, 1979), Lecture Notes in Math., vol. 834, Springer, Berlin, 1980, pp. 381-410. MR 606796 (82m:03045)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 03C40, 14L17, 18D10, 20G05

Retrieve articles in all journals with MSC (2010): 03C40, 14L17, 18D10, 20G05


Additional Information

Moshe Kamensky
Affiliation: Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556
Address at time of publication: Department of Mathematics, Ben-Gurion University, Be’er-Sheva, Israel
Email: kamensky.1@nd.edu, kamenskm@math.bgu.ac.il

DOI: https://doi.org/10.1090/S0002-9947-2014-06062-5
Keywords: Internality, binding group, Galois group, Tannakian formalism, linear algebraic group, linear differential group, representations
Received by editor(s): October 10, 2010
Received by editor(s) in revised form: December 20, 2012
Published electronically: October 10, 2014
Article copyright: © Copyright 2014 American Mathematical Society

American Mathematical Society