The formal theory of relative monads

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 $\mathbb{V}\text{-}\mathbb{C}\mathbf{at}$ of categories enriched in a monoidal category $\mathbb{V}$, though many of our results are new even for $\mathbb{V}=\mathbf{Set}$.

91 pages; v5: amended definition of unitality in a skew-multicategory

References in corpus (1)

• Notes on enriched categories with colimits of some class

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