Abstract
We introduce the notion of a relative pseudomonad, which generalizes the notion of a pseudomonad, and define the Kleisli bicategory associated to a relative pseudomonad. We then present an efficient method to define pseudomonads on the Kleisli bicategory of a relative pseudomonad. The results are applied to define several pseudomonads on the bicategory of profunctors in an homogeneous way and provide a uniform approach to the definition of bicategories that are of interest in operad theory, mathematical logic, and theoretical computer science.
Highlights
Just as classical monad theory provides a general approach to study algebraic structures on objects of a category, 2-dimensional monad theory offers an elegant way to investigate algebraic structures on objects of a 2-category [10,29,34, 37,54,56]
We introduce relative pseudomonads, which generalize pseudomonads, define the the Kleisli bicategory associated to a relative pseudomonad, and describe a method to extend a 2-monad on a 2-category to a pseudomonad on the Kleisli bicategory of a relative pseudomonad
In order to do so, we introduce the notion of a relative pseudomonad (Definition 3.1), which is based on the notions of a relative monad [1, Definition 2.1] and of a noiteration pseudomonad [53, Definition 2.1]
Summary
Just as classical monad theory provides a general approach to study algebraic structures on objects of a category (see [5] for example), 2-dimensional monad theory offers an elegant way to investigate algebraic structures on objects of a 2-category [10,29,34, 37,54,56]. As an illustration of the applications of our theory, we discuss our results in the special case of the 2-monad S for symmetric strict monoidal categories, showing how it can be extended to a pseudomonad on the bicategory of profunctors This result is the cornerstone of the understanding of the bicategory of generalized species of structures defined in [23] as a ‘categorified’ version of the relational model of linear logic [28,30] and leads to a proof that the substitution monoidal structure giving rise to the notion of a coloured operad [4] is a special case of the composition in the Kleisli bicategory.
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