Monoidal bicategories, differential linear logic, and analytic functors (2405.05774v2)

Abstract: We develop further the theory of monoidal bicategories by introducing and studying bicategorical counterparts of the notions of a linear explonential comonad, as considered in the study of linear logic, and of a codereliction transformation, introduced to study differential linear logic via differential categories. As an application, we extend the differential calculus of Joyal's analytic functors to analytic functors between presheaf categories, just as ordinary calculus extends from a single variable to many variables.

• The paper develops bicategorical analogs of structures key to linear logic and extends differential calculus in category theory to analytic functors.
• It introduces a linear exponential pseudocomonad within a bicategorical framework and adapts analytic functor calculus to this higher-dimensional context.
• This research provides theoretical tools for formalizing logic models using bicategorical structures and paves the way for new computational models using bicategorical differential structures.

An Expert Analysis of "Monoidal Bicategories, Differential Linear Logic, and Analytic Functors"

The paper presented here investigates the intricate connections between monoidal bicategories, differential linear logic, and analytic functors by extending two seemingly disparate branches of research. It provides a substantial treatment of monoidal bicategories by developing bicategorical analogs of concepts like linear exponential comonads and codereliction transformations, which are pivotal in linear logic and its differential variant. The paper also extends the scope of differential calculus in category theory to encompass analytic functors amidst presheaf categories.

Key Contributions

1. Monoidal Bicategories and Linear Exponential Structures:
   • The authors introduce bicategorical constructs related to linear exponential comonads—an essential structure utilized in linear logic. A cornerstone of this paper is the introduction of a linear exponential pseudocomonad, providing a bicategorical framework that extends the classical approach from a single dimension (categories) to bidimensional analogs (bicategories).
2. Differential Linear Logic:
   • This work intricately explores the bicategorial structures necessary to support differential linear logic—a logic version endowed with differentiation rules applicable to categories modeling computation. The paper highlights the introduction of differential categories that serve as models for this logic.
3. Analytic Functors:
   • Extending their approach, the authors adapt Joyal's analytic functor calculus into a bicategorical context. This allows a transition from understanding single-variable mathematical calculus to multivariable calculus among categories, offering a robust framework for analyzing complex structuring operations in category theory.

Methodology and Implications

• Pseudodistributivity and Duality:

The research leverages the duality in categories of profunctors to transition from symmetric monoid handling (pseudomonads) to their dual (pseudocomonads), thereby exploring the categorical and bicategorical cores of both logical and analytical frameworks.

• Codification of Linear Logic Models:

By enforcing a systematic investigation on the connections between bicategorical models and logical systems, the paper provides mechanisms to formalize logic models using structural properties derived from the bicategorical view, enhancing methods to represent logical computations.

• Bicategorical Differential Structures:

Through the exposition on codereliction transformations, the paper furnishes bicategorical counterparts to differentiation, paving the way for new computational models that assimilate the nuanced mathematical properties inherent in differential calculus.

Speculations and Future Developments

The implications of this paper extend into several potential future developments:

• Computational Frameworks: The bicategorical view can contribute significantly to structuring complex computations, especially in distributed systems and concurrent computation models.
• Algebraic Geometry and Topological Field Theories: Given the synergy between high-dimensional categories and topological invariants, the methodologies can spawn novel insights into algebraically categorified topological field theories.
• Advanced Logic Systems: By enabling an enriched logical framework encapsulated inside bicategories, there lies a path towards innovating types and reasoning systems in theoretical computer science beyond classic or linear paradigms.

In conclusion, the paper notably contributes to a deeper understanding of the role bicategories play in both logic and analytic functor calculus, offering theoretical tools that extend classical frameworks towards addressing more complex categorical questions. This research opens numerous doors to further inquiry within and beyond its foundational domains.