2023/02/27 by Arkor, Nathanael, McDermott, Dylan · 4 citations
#18A40 #18C15 #18C20 #18D20 #18D60 #18D65 #18D70 #18M50 #18M65 #18N10 #Category Theory (math.CT) #FOS: Mathematics
paper · doi:10.48550/arxiv.2302.14014
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 \mathbbV-ℂat of categories enriched in a monoidal category \mathbbV, though many of our results are new even for \mathbbV = Set.