Monads need not be endofunctors (1412.7148v2)

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.

• The paper introduces relative monads, extending standard monads by using functors between different categories.
• It establishes relative adjunctions and connections to skew-monoidal structures, grounding the theory in robust categorical principles.
• The framework has practical implications in functional programming and computational theory, paving the way for further research.

Relative Monads: A Generalization of Monads Beyond Endofunctors

This paper explores the conceptual expansion of monads within category theory, presenting a framework for what the authors term "relative monads." Unlike traditional monads, which are endofunctors on a single category, relative monads accommodate functors between different categories, broadening their applicability in various computational and theoretical contexts.

Key Concepts and Definitions

The primary contribution of this research is the formal definition and investigation of relative monads. Relative monads extend the notion of monads by allowing for an underlying functor between different categories, rather than restricting to endofunctors. This approach is particularly beneficial in settings where operations are naturally described in terms of functors that do not map a category to itself.

1. Relative Monads: Defined for a functor $J: \mathbb{J} \to \mathbb{C}$, a relative monad consists of a functor $T: |\mathbb{J}| \to |\mathbb{C}|$, a unit natural transformation $\eta$, and a Kleisli extension operation satisfying monadic laws adapted for the relative context.
2. Relative Adjunctions: The paper demonstrates that relative monads correspond to what are termed "relative adjunctions." This is an extension of the conventional adjunction concept, framing relative monads as arising from adjunctions between different categories.
3. Examples and Applications: The authors provide multiple instances of relative monads, including finite-dimensional vector spaces, lambda calculus syntax, and indexed containers. These examples illustrate the versatility of relative monads in modeling computational phenomena beyond the reach of standard monads.

Theoretical Implications

The theoretical implications of this framework are substantial. Relative monads inherit many properties of ordinary monads but offer additional flexibility in terms of categorical constructs. They allow the extension of monads' algebraic properties to settings that involve multiple categories interacting via a functor. Furthermore, the relationship between monads and arrows is reevaluated, presenting arrows as a specific instance of relative monads, countering the previous assumption that arrows generalize monads.

1. Categories of Relative Monads: The authors establish a category of relative monads for a given functor $J$ and demonstrate isomorphisms with categories formed by skew-monoids in skew-monoidal categories. This connection provides a strengthened categorical foundation for relative monads.
2. Well-behavedness Conditions: Essential conditions are identified under which the functor $J$ ensures that relative monads yield properly monoidal structures, allowing for relative monads to extend to full monads in a cocomplete setting.

Practical Applications and Future Work

Relative monads offer intriguing potential for practical applications, especially within the design and analysis of programming languages and in other computational frameworks where categories interface through non-endofunctors. Various practical applications, such as managing side effects in functional programming and optimizing computational structures in quantum computing, could benefit from these theoretical insights.

Future work, as suggested by the authors, includes exploring enriched variants of relative monads and further investigating the structural properties and applications of relative monads in computer science and mathematics. Additionally, formalizing these theoretical constructs in a dependently typed programming language like Agda confirms the computational feasibility and applicability of the principles outlined.

In conclusion, relative monads represent a significant advancement in category theory and functional programming, offering new methods for modeling complex systems across diverse categorical frameworks. These developments pave the way for further exploration in computational theory and practice, offering deeper insights and tools for researchers in these fields.