Monads need not be endofunctors

Abstract

We introduce a generalization of monads, called relative monads, allowing for underlying functors between different categories. Examples include finite-dimensional vector spaces, untyped and typed lambda-calculus syntax and indexed containers. We show that the Kleisli and Eilenberg-Moore constructions carry over to relative monads and are related to relative adjunctions. Under reasonable assumptions, relative monads are monoids in the functor category concerned and extend to monads, giving rise to a coreflection between relative monads and monads. Arrows are also an instance of relative monads.

References in corpus (1)

• Skew monoidales, skew warpings and quantum categories

Cited by in corpus (27)

• Relative pseudomonads, Kleisli bicategories, and substitution monoidal structures
• The formal theory of relative monads
• Enriched algebraic theories and monads for a system of arities
• Fibred Computational Effects
• Proof Theory of Partially Normal Skew Monoidal Categories
• Monoidal computer III: A coalgebraic view of computability and complexity
• Directed Containers as Categories
• Proof Theory of Skew Non-Commutative MILL
• Effects Without Monads: Non-determinism -- Back to the Meta Language
• Semi-Substructural Logics à la Lambek
• Modules over monads and operational semantics (expanded version)
• Relative monadicity
• Semi-Substructural Logics with Additives
• The Quantum Monadology
• Signatures and models for syntax and operational semantics in the presence of variable binding
• The Pebble-Relation Comonad in Finite Model Theory
• Probabilistic Relational Reasoning via Metrics
• Reduction Monads and Their Signatures
• Relating Idioms, Arrows and Monads from Monoidal Adjunctions
• Presentable signatures and initial semantics
• Divergences on Monads for Relational Program Logics
• Equational Reasoning for MTL Type Classes
• What is a monoid?
• Category Theory with Stratified Set Theory
• Fibrational Perspectives on Determinization of Finite-State Automata
• 2-Functoriality of Initial Semantics, and Applications
• On Kleisli liftings and decorated trace semantics