Seeing double through dependent optics

Abstract

Tambara modules are strong profunctors between monoidal categories. They've been defined by Tambara in the context of representation theory [Tam06], 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 [PS08] relevant to profunctor representation theorem for mixed optics [CEG+20] are reobtained in this context. This reproduces the definition of dependent optics recently put forward by Vertechi [Ver22] and Milewski [Mil22], and hinted at in [BCG+21].

Dependent Optics

“…Even though they appear different on the surface, our definition of dependent optics and the notion of compound optics [17] are equivalent (see [4,Ex. 4.2]).…”

“…Here, we will adopt a direct, explicit approach. While the chosen formalisms are different, the two definitions have been shown to be equivalent in [4,Ex. 4.2].…”