Contextads as Wreaths; Kleisli, Para, and Span Constructions as Wreath Products (2410.21889v1)

Abstract: We introduce contextads and the Ctx construction, unifying various structures and constructions in category theory dealing with context and contextful arrows -- comonads and their Kleisli construction, actegories and their Para construction, adequate triples and their Span construction. Contextads are defined in terms of Lack--Street wreaths, suitably categorified for pseudomonads in a tricategory of spans in a 2-category with display maps. The associated wreath product provides the Ctx construction, and by its universal property we conclude trifunctoriality. This abstract approach lets us work up to structure, and thus swiftly prove that, under very mild assumptions, a contextad equipped colaxly with a 2-algebraic structure produces a similarly structured double category of contextful arrows. We also explore the role contextads might play qua dependently graded comonads in organizing contextful computation in functional programming. We show that many side-effects monads can be dually captured by dependently graded comonads, and gesture towards a general result on the `transposability' of parametric right adjoint monads to dependently graded comonads.

• The paper introduces a unified categorical framework by defining contextads through wreath products in tricategories.
• It presents the novel Ctx construction, a double category capturing contextful arrows that unifies Kleisli, Para, and span structures.
• It demonstrates potential applications in programming semantics and system modeling while highlighting the duality of context and computational effects.

Overview of Contextual Structures in Category Theory

The paper by Capucci and Myers explores a unified categorical framework, introducing the notion of "contextads" to elucidate various structures and constructions that deal with context and contextful arrows within category theory. Most notably, this work subsumes the comonads and their general Kleisli constructions, actegories, and graded comonads into a single universal categorical construction termed the "wreath product" in tricategories. Through this abstraction, the authors aim to simplify the portrayal and analysis of parameterized and context-sensitive maps, thus offering a more uniform and abstract understanding of several disparate constructions in higher category theory.

Core Contributions

1. Contextads and Wreath Product: The paper's fundamental contribution is the notion of contextads defined through the framework of tricategories and the use of Lack-Street wreath products. These are conceptualized within the tricategory of spans, which allows for unification of contextual computations across various structures like Kleisli categories and more general span constructions.
2. The $Ctx$ Construction: The authors propose a generalized categorical construction called the $Ctx$ construction. This is essentially a double category that captures contextful arrows arising from contextads, thereby unifying and extending both the Kleisli and Para constructions seen traditionally within categorical constructs. This construction provides a basis for integrating context-sensitive arrows in double categorical settings.
3. Trifunctoriality and Wreath Products: The notion of contextads is deeply entwined with the idea of wreath products in tricategories. The $Ctx$ construction manifests as a trifunctorial operation, derived from such a categorical product, enabling the convenient assembly of various complex structures such as spans within a coherent categorical framework.
4. Application to Systems and Programming: The paper eloquently positions contextads as pivotal structures in addressing context-awareness in functional programming and systems theory. It explores the semantic interpretation of functional programs with context-dependency, underscoring how contextads can manage computational effects and context-awareness comprehensively.
5. Categorification of Context and Effects: An insightful exploration within the paper is the treatment of context and effects, traditionally dual notions in computation. By leveraging dependently graded comonads, the authors show how numerous computational effects, like those captured by polynomial monads, can equivalently be treated via contextads. This opens a door to modeling many-sided computations and systems within this singular abstract framework.

Implications and Future Directions

Through this work, significant implications and future directions for category theory and its applications in computer science and other domains can be delineated:

• Unification and Simplification: By abstracting over varied categorical structures, this work paves a pathway for simplifying the representation of complex systems, making them versatile to new domains such as artificial intelligence and programming languages semantics.
• Extending Applicability: The structured framework set by contextads can be applied for dependencies observed in systems modeling cyber-physical systems, learning paradigms, and more, thereby broadening the applicability of category theory into operational domains.
• Trifunctorial Extensions: The leveraging of trifunctors allows direct utilization upon categories featuring structured dependencies, offering new ways to model evolving and interacting complex systems with context sensitivity deeply embedded.
• Duality and Extendability of Effects and Contexts: The paper notably pushes forward the narrative that effects and contexts, often seen dually, can be modeled uniformly within a contextad framework. This harbors potential for future paper into dualities and extendabilities over a broad set of richly structured mathematical objects.

Conclusion

Capucci and Myers’ work fundamentally enriches the understanding of context within category theory and lays scholarly groundwork for treating context and its dependencies in varied forms. The structured formalism provided by contextads fosters new avenues in both theoretical explorations and practical applications spanning from computer science paradigms to advanced mathematical models in systems theory.