Distributive Laws for Relative Monads - Applied Categorical Structures
- ️Lobbia, Gabriele
- ️Wed Apr 05 2023
Abstract
We introduce the notion of a distributive law between a relative monad and a monad. We call this a relative distributive law and define it in any 2-category \(\mathcal {K}\). In order to do that, we introduce the 2-category of relative monads in a 2-category \(\mathcal {K}\) with relative monad morphisms and relative monad transformations as 1- and 2-cells, respectively. We relate our definition to the 2-category of monads in \(\mathcal {K}\) defined by Street. Using this perspective, we prove two Beck-type theorems regarding relative distributive laws. We also describe what does it mean to have Eilenberg–Moore and Kleisli objects in this context and give examples in the 2-category of locally small categories.
Access this article
Subscribe and save
- Get 10 units per month
- Download Article/Chapter or eBook
- 1 Unit = 1 Article or 1 Chapter
- Cancel anytime
Buy Now
Price excludes VAT (USA)
Tax calculation will be finalised during checkout.
Instant access to the full article PDF.
Similar content being viewed by others
Data Availability
Data sharing not applicable to this article as no datasets were generated or analysed during the current study.
References
Aczel, P.: The type theoretic interpretation of constructive set theory. Volume 77 of Logic Colloquium, pp. 55–66, North-Holland (1978)
Ahrens, B.: Initiality for typed syntax and semantics. In: Ong, L., de Queiroz, R. (eds.) Logic. Language, Information and Computation, pp. 127–141. Berlin, Heidelberg (2012)
Ahrens, B.: Modules over relative monads for syntax and semantics. Math. Struct. Comput. Sci. 26(1), 3–37 (2016)
Altenkirch, T., Chapman, J., Uustalu, T.: Monads need not to be endofunctors. Log. Methods Comput. Sci. 11(1:3), 1–40 (2015)
Arkor, N.: Monadic and Higher-Order Structure. PhD thesis, University of Cambridge, (2022)
Barr, M., Wells, C.: Toposes, Triples and Theories. Springer (1985)
Barwise, J.: Admissible Sets and Structures. Cambridge University Press (2017)
Beck, J.: Distributive laws. In Seminar on Triples and Categorical Homology Theory, volume 80 of Lecture Notes in Mathematics. Springer (1969)
Blackwell, R., Kelly, G.M., Power, A.J.: Two-dimensional monad theory. J. Pure Appl. Algebra 59(1), 1–41 (1989)
Bourke, J.: Codescent objects in 2-dimensional universal algebra. PhD thesis, University of Sidney, (2010)
Fiore, M., Gambino, N., Hyland, M., Winskel, G.: Relative pseudomonads, kleisli bicategories, and substitution monoidal structures. Selecta Mathematica (2017)
Gray, J.W.: Formal category theory: Adjointness for 2-categories. In Lecture Notes in Mathematics, volume 391 (1974)
Hernández, E.R.: Another characterization of no-iteration distributive laws. preprint, arXiv:1910.06531v2 (2020)
Hyland, M.: Some reasons for generalising domain theory. Math. Struct. Comput. Sci. 20(2), 239–265 (2010)
Kelly, G.M.: Basic Concepts of Enriched Category Theory. Cambridge University Press (1982)
Lack, S.: A 2-Categories Companion. Springer, New York, pp. 105–191 (2010)
Lack, S., Street, R.: The formal theory of monads II. J. Pure Appl. Algebra 175(1–3), 243–265 (2002)
Lawvere, F. W.: Functorial Semantics of Algebraic Theories and Some Algebraic Problems in the context of Functorial Semantics of Algebraic Theories. PhD thesis, Columbia University, (1963)
Manes, E.G.: Algebraic Theories, volume 26 of Graduate Texts in Mathematics. Springer (1976)
Manes, E.G., Mulry, P.: Monad compositions I: general constructions and recursive distributive laws. Theory Appl. Categ. 18(7), 172–208 (2007)
Marmolejo, F.: Distributive laws for pseudomonads. Theory and Applications of Categories, 5, (1999)
Marmolejo, F., Vázquez-Márquez, A.: No-iteration mixed distributive laws. Math. Struct. Comput. Sci. 27, 1–16 (2017)
Marmolejo, F., Wood, R.J.: Monads as extensions systems - no iteration is necessary. Theory Appl. Categ. 24(4), 84–113 (2010)
Marmolejo, F., Rosebrugh, R.D., Wood, R.J.: A basic distributive law. J. Pure Appl. Algebra 168(2–3), 209–226 (2002)
Moggi, E.: Notions of computation and monads. Inf. Comput. 93(1), 55–92 (1991)
Power, A.J.: A 2-categorical pasting theorem. J. Algebra 129, 439–445 (1990)
Staton, S.: An algebraic presentation of predicate logic. In: Pfenning, F. (ed.), Foundations of software science and computation structures, pp. 401–417, Berlin, Heidelberg (2013)
Street, R.: Fibrations and Yoneda’s lemma in a 2-category. In: Kelly, G. M. (ed.) Category Seminar, pp. 104–133. Berlin, Heidelberg (1974)
Street, R.: The formal theory of monads. J. Pure Appl. Algebra 2(2), 149–168 (1972)
Street, R., Walters, R.: Yoneda structures on 2-categories. J. Algebra 50, 350–379 (1978)
Ulmer, F.: Properties of dense and relative adjoint functors. J. Algebra 8(1), 77–95 (1968)
Weber, M.: Yoneda structures from 2-toposes. Appl. Categ. Struct. 15, 259–323 (2007)
Acknowledgements
The author is very grateful to Martin Hyland for the helpful conversation about the definition of extension to Kleisli in this particular case. This paper also owes a lot to Nicola Gambino’s suggestions and feedback. Discussions with Francesco Gallinaro and Giovanni Soldà helped the author dealing with some examples. This research is part of the author’s PhD project, supported by an EPSRC Scholarship. The second version of this paper was improved also thanks to suggestions and comments by Nathanael Arkor and John Bourke.
Funding
This research is part of the author’s PhD project, supported by an EPSRC Scholarship.
Author information
Authors and Affiliations
School of Mathematics, University of Leeds, Leeds, UK
Gabriele Lobbia
Authors
- Gabriele Lobbia
You can also search for this author in PubMed Google Scholar
Corresponding author
Correspondence to Gabriele Lobbia.
Ethics declarations
Conflict of interest
The author declares they have no competing interests.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Communicated by Ross Street.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Lobbia, G. Distributive Laws for Relative Monads. Appl Categor Struct 31, 19 (2023). https://doi.org/10.1007/s10485-023-09716-1
Received: 01 July 2022
Accepted: 23 February 2023
Published: 05 April 2023
DOI: https://doi.org/10.1007/s10485-023-09716-1