Relative pseudomonads, Kleisli bicategories, and substitution monoidal structures

Abstract

We introduce the notion of a relative pseudomonad, which generalises the notion of a pseudomonad, and define the Kleisli bicategory associated to a relative pseudomonad. We then present an efficient method to define pseudomonas 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, thus providing a uniform approach to the definition of bicategories that are of interest in operad theory, mathematical logic, and theoretical computer science.

v3: Following referee comments: some material reorganised, presentation streamlined, definition of lax idempotent relative pseudomonad rephrased in terms of extensions. 32 pages. Accepted for publication in Selecta Mathematica (New Series)

References in corpus (3)

• Lax formal theory of monads, monoidal approach to bicategorical structures and generalized operads
• Operads, tensor products, and the categorical Borel construction
• Constructing Fully Complete Models of Multiplicative Linear Logic

Cited by in corpus (13)

• Coend calculus
• The formal theory of relative monads
• Accessibility and presentability in 2-categories
• Free Commutative Monoids in Homotopy Type Theory
• A type theory for cartesian closed bicategories
• Distributive laws for relative monads
• The linear-non-linear substitution 2-monad
• A unified framework for notions of algebraic theory
• Foundations of Algebraic Theories and Higher Dimensional Categories
• Bicategories of Automata, Automata in Bicategories
• Optics for Premonoidal Categories
• Presentations of pseudodistributive laws
• Stabilized profunctors and stable species of structures