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Categorical entropy, (co-)t-structures and ST-triples

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Abstract

In this paper, we study a dynamical property of an exact endofunctor \(\Phi : {\mathcal {D}} \rightarrow {\mathcal {D}}\) of a triangulated category \({\mathcal {D}}\). In particular, we are interested in the following question: Given full triangulated subcategories \({\mathcal {A}},{\mathcal {B}} \subset {\mathcal {D}}\) such that \(\Phi ({\mathcal {A}}) \subset {\mathcal {A}}\) and \(\Phi ({\mathcal {B}}) \subset {\mathcal {B}}\), how the categorical entropies of \(\Phi |_{\mathcal {A}}\) and \(\Phi |_{\mathcal {B}}\) are related? To answer this question, we introduce new entropy-type invariants using bounded (co-)t-structures with finite (co-)hearts and prove their basic properties. We then apply these results to answer our question for the situation where \({\mathcal {A}}\) has a bounded t-structure and \({\mathcal {B}}\) has a bounded co-t-structure which are, in some sense, dual to each other.

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The data that support the findings of this study are available from the author upon reasonable request.

References

  1. Abouzaid, M., Seidel, P.: An open string analogue of Viterbo functoriality. Geom. Topol. 14, 627–718 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  2. Adachi, T., Mizuno, Y., Yang, D.: Discreteness of silting objects and t-structures in triangulated categories. Proc. Lond. Math. Soc. 118, 1–42 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  3. Amiot, C.: Cluster categories for algebras of global dimension 2 and quivers with potential. Ann. Inst. Fourier (Grenoble) 59, 2525–2590 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  4. Assem, I., Simson, D., Skowroński, A.: Elements of the Representation Theory of Associative Algebras. Volume 1: Techniques of Representation Theory, London Mathematical Society Student Texts, vol. 65, Cambridge University Press, Cambridge (2006)

  5. Bae, H., Jeong, W., Kim, J.: Cluster categories from Fukaya categories, arXiv:2209.09442

  6. Beilinson, A., Bernstein, J., Deligne, P.: Faisceaux pervers. Astérisque 100, 5–171 (1983)

  7. Bondal, A., Kapranov, M.: Representable functors, Serre functors, and mutations. Math. USSR Izv. 35, 519–541 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bondarko, M.V.: Weight structures vs. t-structures; weight filtrations, spectral sequences, and complexes (for motives and in general). J. K-theory 6, 387–504 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  9. Dimitrov, G., Haiden, F., Katzarkov, L., Kontsevich, M.: Dynamical systems and categories. Contemp. Math. 621, 133–170 (2014). (AMS)

    Article  MathSciNet  MATH  Google Scholar 

  10. Etgü, T., Lekili, Y.: Koszul duality patterns in Floer theory. Geom. Topol. 21, 3313–3389 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  11. Fan, Y.-W., Fu, L., Ouchi, G.: Categorical polynomial entropy. Adv. Math. 383, 107655 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  12. Ginzburg, V.: Calabi–Yau algebras. arXiv:math/0612139

  13. Gromov, M.: Entropy, homology and semialgebraic geometry. Astérisque 145–146, 225–240 (1987)

    MathSciNet  MATH  Google Scholar 

  14. Hochenegger, A., Kalck, M., Ploog, D.: Spherical subcategories in algebraic geometry. Math. Nachr. 289, 1450–1465 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  15. Ikeda, A.: Mass growth of objects and categorical entropy. Nagoya Math. J. 244, 136–157 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  16. Iyama, O., Yang, D.: Silting reduction and Calabi-Yau reduction of triangulated categories. Trans. Am. Math. Soc. 370, 7861–7898 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  17. Jørgensen, P.: Co-t-structures: The first decade. In: Surveys in Representation Theory of Algebra, Contemporary Mathematics, vol. 716, pp. 25–36. American Mathematical Society, Providence, RI (2018)

  18. Kalck, M., Yang, D.: Relative singularity categories I: Auslander resolutions. Adv. Math. 301, 973–1021 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  19. Keller, B.: Deriving DG categories. Ann. Sci. École Norm. Sup. 27, 63–102 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  20. Keller, B.: Calabi–Yau triangulated categories. In: Trends in Representation Theory of Algebras and Related Topics, EMS Series of Congress Reports, pp. 467–489. European Mathematical Society (2008)

  21. Keller, B., Yang, D.: Derived equivalences from mutations of quivers with potential. Adv. Math. 226, 2118–2168 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  22. Le, J., Chen, X.-W.: Karoubianness of a triangulated category. J. Algebra 310, 452–457 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  23. Orlov, D.: Derived categories of coherent sheaves and triangulated categories of singularities. In: Algebra, Arithmetic, and Geometry, Progress in Mathematics, vol. 270, pp. 503–531. Birkhäuser, Boston, MA (2009)

  24. Pauksztello, D.: Compact corigid objects in triangulated categories and co-t-structures. Cent. Eur. J. Math. 6, 25–42 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  25. Seidel, P.: Fukaya categories and Picard–Lefschetz theory. In: Zurich Lectures in Advanced Mathematics. European Mathematical Society, Zürich (2008)

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Acknowledgements

This work was supported by the Institute for Basic Science (IBS-R003-D1).

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  1. Center for Geometry and Physics, Institute for Basic Science (IBS), Pohang, 37673, Republic of Korea

    Jongmyeong Kim

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  1. Jongmyeong Kim
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Kim, J. Categorical entropy, (co-)t-structures and ST-triples. Math. Z. 304, 37 (2023). https://doi.org/10.1007/s00209-023-03293-8

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