Model theory and the Tannakian formalism

Kamensky, Moshe

doi:10.1090/S0002-9947-2014-06062-5

Model theory and the Tannakian formalism
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Trans. Amer. Math. Soc. 367 (2015), 1095-1120 Request permission

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

P. J. Cassidy, Differential algebraic groups, Amer. J. Math. 94 (1972), 891–954. MR 360611, DOI 10.2307/2373764
Phyllis Joan Cassidy, The differential rational representation algebra on a linear differential algebraic group, J. Algebra 37 (1975), no. 2, 223–238. MR 409426, DOI 10.1016/0021-8693(75)90075-7
P. Deligne, Catégories tannakiennes, The Grothendieck Festschrift, Vol. II, Progr. Math., vol. 87, Birkhäuser Boston, Boston, MA, 1990, pp. 111–195 (French). MR 1106898
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-New York, 1982. MR 654325
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
Revêtements étales et groupe fondamental (SGA 1), Documents Mathématiques (Paris) [Mathematical Documents (Paris)], vol. 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
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, DOI 10.2178/jsl/1120224718
Ehud Hrushovski, Computing the Galois group of a linear differential equation, Differential Galois theory (Będlewo, 2001) Banach Center Publ., vol. 58, Polish Acad. Sci. Inst. Math., Warsaw, 2002, pp. 97–138. MR 1972449, DOI 10.4064/bc58-0-9
Ehud Hrushovski, Groupoids, imaginaries and internal covers, Turkish J. Math. 36 (2012), no. 2, 173–198. MR 2912035, DOI 10.3906/mat-1001-91
Ehud Hrushovski, Unidimensional theories are superstable, Ann. Pure Appl. Logic 50 (1990), no. 2, 117–138. MR 1081816, DOI 10.1016/0168-0072(90)90046-5
Bradd Hart, Thomas G. Kucera, Anand Pillay, Philip J. Scott, and Robert A. G. Seely (eds.), 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. MR 2848606, DOI 10.1090/crmp/053
Moshe Kamensky, Tannakian formalism over fields with operators, Int. Math. Res. Not. IMRN 24 (2013), 5571–5622. MR 3144174, DOI 10.1093/imrn/rns190
E. R. Kolchin, Differential algebra and algebraic groups, Pure and Applied Mathematics, Vol. 54, Academic Press, New York-London, 1973. MR 0568864
Saunders Mac Lane, Categories for the working mathematician, 2nd ed., Graduate Texts in Mathematics, vol. 5, Springer-Verlag, New York, 1998. MR 1712872
David Marker, Model theory, Graduate Texts in Mathematics, vol. 217, Springer-Verlag, New York, 2002. An introduction. MR 1924282
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
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; A K Peters, Ltd., Wellesley, MA, 2006. MR 2215060
James S. Milne, Étale cohomology, Princeton Mathematical Series, No. 33, Princeton University Press, Princeton, N.J., 1980. MR 559531
Alexey Ovchinnikov, Tannakian approach to linear differential algebraic groups, Transform. Groups 13 (2008), no. 2, 413–446. MR 2426137, DOI 10.1007/s00031-008-9010-4
Alexey Ovchinnikov, Tannakian categories, linear differential algebraic groups, and parametrized linear differential equations, Transform. Groups 14 (2009), no. 1, 195–223. MR 2480859, DOI 10.1007/s00031-008-9042-9
Anand Pillay, Algebraic $D$-groups and differential Galois theory, Pacific J. Math. 216 (2004), no. 2, 343–360. MR 2094550, DOI 10.2140/pjm.2004.216.343
A. Pillay, Model theory, Course lecture notes. 2002. URL: http://www.math.uiuc.$\space$edu/People/pillay/lecturenotes_modeltheory.pdf (cit. on pp. 3, 4).
Anand Pillay, Some foundational questions concerning differential algebraic groups, Pacific J. Math. 179 (1997), no. 1, 179–200. MR 1452531, DOI 10.2140/pjm.1997.179.179
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, DOI 10.1090/surv/087
Bruno Poizat, Une théorie de Galois imaginaire, J. Symbolic Logic 48 (1983), no. 4, 1151–1170 (1984) (French). MR 727805, DOI 10.2307/2273680
Neantro Saavedra Rivano, Catégories Tannakiennes, Lecture Notes in Mathematics, Vol. 265, Springer-Verlag, Berlin-New York, 1972 (French). MR 0338002
William C. Waterhouse, Introduction to affine group schemes, Graduate Texts in Mathematics, vol. 66, Springer-Verlag, New York-Berlin, 1979. MR 547117
B. I. Zil′ber, 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