[Paper Review] Seeing double through dependent optics

Matteo Capucci|arXiv (Cornell University)|Apr 22, 2022

Logic, programming, and type systems4 citations

TL;DR

This paper extends Tambara module theory to actions of double categories (doubly indexed categories), defining Tambara modules as horizontal natural transformations. It recovers recent definitions of dependent optics—such as those by Vertechi and Milewski—via a profunctor representation theorem, unifying mixed optics and dependent lenses in a higher-dimensional categorical framework.

ABSTRACT

Tambara modules are strong profunctors between monoidal categories. They've been defined by Tambara in the context of representation theory, but quickly found their way in applications when it was understood Tambara modules provide a useful encoding of modular data accessors known as mixed optics. To suit the needs of these applications, Tambara theory has been extended to profunctors between categories receiving an action of a monoidal category. Motivated by the generalization of optics to dependently-typed contexts, we sketch a further extension of the theory of Tambara modules in the setting of actions of double categories (thus doubly indexed categories), by defining them as horizontal natural transformations. The theorems and constructions in Pastro-Street theory relevant to profunctor representation theorem for mixed optics are reobtained in this context. This reproduces the definition of dependent optics recently put forward by Vertechi and Milewski, and hinted at by previous work of the author and his collaborators.

Motivation & Objective

• To generalize Tambara modules to actions of double categories, enabling a unified framework for dependent optics.
• To provide a categorical foundation for dependent optics that subsumes prior constructions like indexed optics and ommatidia.
• To recover the profunctor representation theorem for mixed optics in the context of doubly indexed categories.
• To establish a connection between double category theory and the semantics of bidirectional data accessors in dependent type theory.
• To demonstrate that the proposed framework captures known examples of dependent optics, such as those defined by Vertechi and Milewski.

Proposed method

• Define Tambara modules as horizontal natural transformations between doubly indexed categories (actions of double categories).
• Use the structure of double categories to model parameterized morphisms and pullbacks in optics, generalizing the standard coend construction.
• Reformulate the profunctor representation theorem from [PS08] in the context of actions on doubly indexed categories.
• Apply the framework to recover the definition of dependent optics as horizontal natural transformations, generalizing both lenses and mixed optics.
• Leverage the Yoneda theory for horizontal natural transformations to derive representation results.
• Show that the construction generalizes existing definitions by Vertechi and Milewski by embedding their examples into the double-categorical framework.

Experimental results

Research questions

• RQ1 How can Tambara modules be generalized to actions of double categories to model dependent optics?
• RQ2 Can the profunctor representation theorem for mixed optics be recovered in a doubly indexed categorical setting?
• RQ3 What is the relationship between horizontal natural transformations and the semantics of dependent optics?
• RQ4 Which examples of dependent optics are captured by the double-categorical framework that are not captured by bicategorical or 2-categorical approaches?
• RQ5 How does the double-categorical structure refine or complete the existing theory of optics beyond framed bicategories?

Key findings

• The paper defines Tambara modules as horizontal natural transformations between doubly indexed categories, providing a clean and general definition of dependent optics.
• The framework recovers the profunctor representation theorem for mixed optics in the context of double categories, extending the result from [PS08] to this new setting.
• The construction reproduces the definition of dependent optics recently proposed by Vertechi [Ver22] and Milewski [Mil22], validating the approach.
• The theory captures known examples of dependent optics, including those based on coproducts and indexed morphisms, through the lens of double-categorical actions.
• The use of horizontal natural transformations provides a principled and structurally sound foundation for dependent optics, aligning with Yoneda-style reasoning.
• The framework suggests that double category theory may offer a more complete and coherent foundation for optics than previous bicategorical or 2-categorical approaches, especially in capturing feedback and side-effect structures.