Document Open Access Logo

Decomposing Comonad Morphisms

Authors Danel Ahman , Tarmo Uustalu

PDF

File

Thumbnail PDF
PDF
LIPIcs.CALCO.2019.14.pdf
  • Filesize: 0.55 MB
  • 19 pages

Document Identifiers

Subject Classification

ACM Subject Classification
  • Theory of computation → Logic
  • Theory of computation → Categorical semantics
Keywords
  • container functors (polynomial functors)
  • container comonads
  • comonad morphisms and comonad coalgebras
  • cofunctors
  • lenses

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Abstract

The analysis of set comonads whose underlying functor is a container functor in terms of directed containers makes it a simple observation that any morphism between two such comonads factors through a third one by two comonad morphisms, whereof the first is identity on shapes and the second is identity on positions in every shape. This observation turns out to generalize into a much more involved result about comonad morphisms to comonads whose underlying functor preserves Cartesian natural transformations to itself on any category with finite limits. The bijection between comonad coalgebras and comonad morphisms from costate comonads thus also yields a decomposition of comonad coalgebras.

Cite As Get BibTex

Danel Ahman and Tarmo Uustalu. Decomposing Comonad Morphisms. In 8th Conference on Algebra and Coalgebra in Computer Science (CALCO 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 139, pp. 14:1-14:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019) https://doi.org/10.4230/LIPIcs.CALCO.2019.14

Author Details

Danel Ahman
  • Faculty of Physics and Mathematics, University of Ljubljana, Slovenia
Tarmo Uustalu
  • Department of Computer Science, Reykjavik University, Iceland
  • Dept. of Software Science, Tallinn University of Technology, Estonia

Funding

  • Ahman, Danel: This material is based upon work supported by the Air Force Office of Scientific Research under award number FA9550-17-1-0326.
  • Uustalu, Tarmo: Supported by the Icelandic Research Fund project grant no. 196323-051 and the Estonian Ministry of Education and Research institutional research grant no. IUT33-13.

Acknowledgements

We thank our anonymous reviewers for very useful remarks.

References

  1. Michael Abbott, Thorsten Altenkirch, and Neil Ghani. Containers: Constructing Strictly Positive Types. Theor. Comput. Sci., 342(1):3-27, 2005. URL: https://doi.org/10.1016/j.tcs.2005.06.002.
  2. Marcelo Aguiar. Internal Categories and Quantum Groups. PhD thesis, Cornell University, 1997. URL: http://www.math.cornell.edu/~maguiar/thesis2.pdf.
  3. Danel Ahman, James Chapman, and Tarmo Uustalu. When Is a Container a Comonad? Log. Methods Comput. Sci., 10(3), 2014. article 14. URL: https://doi.org/10.2168/lmcs-10(3:14)2014.
  4. Danel Ahman and Tarmo Uustalu. Distributive Laws of Directed Containers. Progress in Informatics, 10:3-18, 2013. URL: https://doi.org/10.2201/niipi.2013.10.2.
  5. Danel Ahman and Tarmo Uustalu. Directed Containers as Categories. In Robert Atkey and Neil Krishnaswami, editors, Proc. of 6th Wksh. on Mathematically Structured Functional Programming, MSFP 2016 (Eindhoven, Apr. 2016), volume 207 of Electron. Proc. in Theor. Comput. Sci., pages 89-98. Open Publishing Assoc., 2016. URL: https://doi.org/10.4204/eptcs.207.5.
  6. Danel Ahman and Tarmo Uustalu. Taking Updates Seriously. In Romina Eramo and Michael Johnson, editors, Proc. of 6th Wksh. on Mathematically Structured Functional Programming, MSFP 2016 (Eindhoven, Apr. 2016), volume 1827 of CEUR Wksh. Proc., pages 59-73. RWTH Aachen, 2017. URL: http://ceur-ws.org/Vol-1827/paper11.pdf.
  7. Nicola Gambino and Martin Hyland. Wellfounded Trees and Dependent Polynomial Functors. In Stefano Berardi, Mario Coppo, and Ferruccio Damiani, editors, Revised Selected Papers from Int. Wksh. on Types for Proofs and Programs, TYPES 2003 (Torino, Apr./May 2003), volume 3085 of Lect. Notes in Comput. Sci., pages 210-225. Springer, 2004. URL: https://doi.org/10.1007/978-3-540-24849-1_14.
  8. Nicola Gambino and Joachim Kock. Polynomial Functors and Polynomial Monads. Math. Proc. Cambridge Philos. Soc., 154(1):153-192, 2013. URL: https://doi.org/10.1017/s0305004112000394.
  9. Richard Garner. Double Clubs. Cahiers de Topologie et Géométrie Différentielle Catégoriques,, 47(4):261-316, 2006. URL: http://www.numdam.org/item?id=CTGDC_2006__47_4_261_0.
  10. Martin Hyland, Paul B. Levy, Gordon Plotkin, and John Power. Combining Algebraic Effects with Continuations. Theor. Comput. Sci., 375(1-3):20-40, 2007. URL: https://doi.org/10.1016/j.tcs.2006.12.026.
  11. C. Barry Jay. A Semantics for Shape. Sci. Comput. Program., 25(2-3):251-283, 1995. URL: https://doi.org/10.1016/0167-6423(95)00015-1.
  12. C. Barry Jay and J. Robin B. Cockett. Shapely Types and Shape Polymorphism. In Donald Sannella, editor, Proc. of 5th European Symp. on Programming Languages and Systems, ESOP '94 (Edinburgh, Apr. 1994), volume 788 of Lect. Notes in Comput. Sci., pages 302-316. Springer, 2004. URL: https://doi.org/10.1007/3-540-57880-3_20.
  13. G. Max Kelly. A Unified Treatment of Transfinite Constructions for Free Algebras, Free monoids, Colimits, Associated Sheaves, and So on. Bull. Austral. Math. Soc., 22(1):1-83, 1980. URL: https://doi.org/10.1017/s0004972700006353.
  14. G. Max Kelly. On Clubs and Data-Type Constructors. In Michael P. Fourman, Peter T. Johnstone, and Andrew M. Pitts, editors, Applications of Categories in Computer Science, volume 177 of London. Math. Soc. Lect. Note Series, pages 163-190. Cambridge Univ. Press, 1992. URL: https://doi.org/10.1017/cbo9780511525902.010.
  15. nLab authors. Full Image. nLab entry, revision 5, April 2016. URL: http://ncatlab.org/nlab/revision/full%20image/5.
  16. John Power and Olha Shkaravska. From Comodels to Coalgebras: State and Arrays. Electron. Notes Theor. Comput. Sci., 106:297-314, 2004. URL: https://doi.org/10.1016/j.entcs.2004.02.041.
  17. Tarmo Uustalu. Container Combinatorics: Monads and Lax Monoidal Functors. In Mohammad Reza Mousavi and Jiří Sgall, editors, Proc. of 2nd IFIP WG 1.8 Int. Conf. on Topics in Theoretical Computer Science, TTCS 2017 (Tehran, Sept. 2017), volume 10608 of Lect. Notes in Comput. Sci., pages 91-105. Springer, 2017. URL: https://doi.org/10.1007/978-3-319-68953-1_8.
  18. Mark Weber. Polynomials in Categories with Pullbacks. Theor. Appl. Categ., 30(16):533-598, 2015. URL: http://www.tac.mta.ca/tac/volumes/30/16/30-16abs.html.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact