2022/04/22 by Matteo Capucci, Capucci, Matteo · 4 citations
Computer Science · Decision Sciences · Mathematics · #Algebra over a field #Algorithm #Computer science #Context (archaeology) #Generalization #Homotopy and Cohomology in Algebraic Topology #Logic, programming, and type systems #Mathematical analysis #Mathematics #Modular design #Programming language #Pure mathematics #Representation (politics) #Scientific Computing and Data Management #Sketch #math.CT
paper · pdf · doi:10.48550/arxiv.2204.10708
published in arXiv (Cornell University) (Cornell University) · 17 pages, 1 figure
arxiv created 2022/04/22 · openalex publication_date 2022/04/22 · arxiv updated 2022/04/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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.