Clones from comonoids

• Ulrich Krähmer ^[1] ; Myriam Mahaman ^[1]
  1. [1] TU Dresden, Institute of Geometry, Germany
• Localización: Revista de la Unión Matemática Argentina, ISSN 0041-6932, ISSN-e 1669-9637, Vol. 68, Nº. 2, 2025, págs. 369-394
• Idioma: inglés
• DOI: 10.33044/revuma.3951
• Enlaces
  • Texto completo
• Resumen

  • We revisit the fact that the cocommutative comonoids in a symmetric monoidal category form the best possible approximation by a cartesian category, now considering the case where the original category is only braided monoidal. This leads to the question of when the endomorphism operad of a comonoid is a clone (a Lawvere theory). By giving an explicit example, we prove that this does not imply that the comonoid is cocommutative.

• Referencias bibliográficas
  • A. Akhvlediani, Relating types in categorical universal algebra, Ph.D. thesis, University of Oxford, 2011.
  • J. C. Baez, Hochschild homology in a braided tensor category, Trans. Amer. Math. Soc. 344 no. 2 (1994), 885–906. DOI MR Zbl
  • R. Blackwell, G. M. Kelly, and A. J. Power, Two-dimensional monad theory, J. Pure Appl. Algebra 59 no. 1 (1989), 1–41. DOI MR Zbl
  • P.-L. Curien, Operads, clones, and distributive laws, in Operads and universal algebra, Nankai Ser. Pure Appl. Math. Theoret. Phys. 9, World...
  • T. Fox, Coalgebras and Cartesian categories, Comm. Algebra 4 no. 7 (1976), 665–667. DOI MR Zbl
  • M. R. Gould, Coherence for categorified operadic theories, Ph.D. thesis, University of Glasgow, 2008. Available at https://theses.gla.ac.uk/id/eprint/689.
  • R. Hartshorne, Algebraic geometry, Graduate Texts in Mathematics 52, Springer, New York-Heidelberg, 1977. MR Zbl
  • C. Heunen and J. Vicary, Categories for quantum theory: An introduction, Oxford Graduate Texts in Mathematics 28, Oxford University Press,...
  • M. Hyland, Towards a notion of lambda monoid, in Proceedings of the Workshop on Algebra, Coalgebra and Topology (WACT 2013), Electron. Notes...
  • G. M. Kelly, An abstract approach to coherence, in Coherence in categories, Lecture Notes in Math. 281, Springer, Berlin-New York, 1972, pp....
  • G. M. Kelly, Many-variable functorial calculus. I, in Coherence in categories, Lecture Notes in Math. 281, Springer, Berlin-New York, 1972,...
  • S. Kerkhoff, R. Poschel ¨ , and F. M. Schneider, A short introduction to clones, in Proceedings of the Workshop on Algebra, Coalgebra and...
  • F. W. Lawvere, Functorial semantics of algebraic theories and some algebraic problems in the context of functorial semantics of algebraic...
  • T. Leinster, Higher operads, higher categories, London Math. Soc. Lecture Note Ser. 298, Cambridge University Press, Cambridge, 2004. DOI...
  • J.-L. Loday, Cyclic homology, second ed., Grundlehren Math. Wiss. 301, Springer, Berlin, 1998. DOI MR Zbl
  • J.-L. Loday and B. Vallette, Algebraic operads, Grundlehren Math. Wiss. 346, Springer, Heidelberg, 2012. DOI MR Zbl
  • M. Shulman, Categorical logic from a categorical point of view, 2016. Available at https: //mikeshulman.github.io/catlog/catlog.pdf, accessed...
  • M. Shulman, P. L. Lumsdaine, and J. A. Gross, Notes on higher categories and categorical logic, lecture notes, August 2016. Available at https://docslib.org/doc/8484804/ notes-on-higher-categories-and-categorical-logic,...
  • S. N. Tronin, Abstract clones and operads, Sibirsk. Mat. Zh. 43 no. 4 (2002), 924–936; translation in Siberian Math. J. 43 no. 4 (2002), 746–755....
  • C. A. Weibel, An introduction to homological algebra, Cambridge Stud. Adv. Math. 38, Cambridge University Press, Cambridge, 1994. DOI MR Zbl