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 (9)

• Higher-dimensional Algebra and Topological Quantum Field Theory
• A 2-categories companion
• Limits of small functors
• Monads need not be endofunctors
• On operads, bimodules and analytic functors
• A unified framework for generalized multicategories
• 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

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• 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