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.

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Cited by in corpus (29)

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