Publication details
The formal theory of relative monads
| Authors | |
|---|---|
| Year of publication | 2024 |
| Type | Article in Periodical |
| Magazine / Source | Journal of Pure and Applied Algebra |
| MU Faculty or unit | |
| Citation | |
| Web | https://www.sciencedirect.com/science/article/pii/S0022404924000732 |
| Doi | http://dx.doi.org/10.1016/j.jpaa.2024.107676 |
| Keywords | Enriched category theory; Formal category theory; Relative adjunction; Relative monad; Skew-monoidal category; Virtual equipment |
| Description | 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. |