ARKOR, Nathanael Amariah and Dylan MCDERMOTT. The formal theory of relative monads. Journal of Pure and Applied Algebra. Elsevier B.V., 2024, vol. 228, No 9, p. 1-107. ISSN 0022-4049. Available from: https://dx.doi.org/10.1016/j.jpaa.2024.107676.
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Basic information
Original name The formal theory of relative monads
Authors ARKOR, Nathanael Amariah (826 United Kingdom of Great Britain and Northern Ireland, belonging to the institution) and Dylan MCDERMOTT.
Edition Journal of Pure and Applied Algebra, Elsevier B.V. 2024, 0022-4049.
Other information
Original language English
Type of outcome Article in a journal
Field of Study 10101 Pure mathematics
Country of publisher Netherlands
Confidentiality degree is not subject to a state or trade secret
WWW URL
Impact factor Impact factor: 0.800 in 2022
Organization unit Faculty of Science
Doi http://dx.doi.org/10.1016/j.jpaa.2024.107676
UT WoS 001223882300001
Keywords in English Enriched category theory; Formal category theory; Relative adjunction; Relative monad; Skew-monoidal category; Virtual equipment
Tags rivok
Tags International impact, Reviewed
Changed by Changed by: Mgr. Marie Šípková, DiS., učo 437722. Changed: 11/10/2024 13:57.
Abstract
We develop the theory of relative monads and relative adjunctions in a virtual equipment, extending the theory of monads and adjunctions in a 2-category. The theory of relative comonads and relative coadjunctions follows by duality. While some aspects of the theory behave analogously to the non-relative setting, others require new insights. In particular, the universal properties that define the algebra object and the opalgebra object for a monad in a virtual equipment are stronger than the classical notions of algebra object and opalgebra object for a monad in a 2-category. Inter alia, we prove a number of representation theorems for relative monads, establishing the unity of several concepts in the literature, including the devices of Walters, the j-monads of Diers, and the relative monads of Altenkirch, Chapman, and Uustalu. A motivating setting is the virtual equipment V-Cat of categories enriched in a monoidal category V, though many of our results are new even for V = Set.
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