Dependent Optics

Categories of Optics

Bidirectional data accessors such as lenses, prisms and traversals are all instances of the same general 'optic' construction. We give a careful account of this construction and show that it extends to a functor from the category of symmetric monoidal categories to itself. We also show that this construction enjoys a universal property: it freely adds counit morphisms to a symmetric monoidal category. Missing in the folklore is a general definition of 'lawfulness' that applies directly to any optic category. We provide such a definition and show that it is equivalent to the folklore profunctor optic laws.

Profunctor Optics, a Categorical Update

Optics are bidirectional data accessors that capture data transformation patterns such as accessing subfields or iterating over containers. Profunctor optics are a particular choice of representation supporting modularity, meaning that we can construct accessors for complex structures by combining simpler ones. Profunctor optics have previously been studied only in an unenriched and non-mixed setting, in which both directions of access are modelled in the same category. However, functional programming languages are arguably better described by enriched categories; and we have found that some structures in the literature are actually mixed optics, with access directions modelled in different categories. Our work generalizes a classic result by Pastro and Street on Tambara theory and uses it to describe mixed V-enriched profunctor optics and to endow them with V-category structure. We provide some original families of optics and derivations, including an elementary one for traversals. Finally, we discuss a Haskell implementation.

What you needa know about Yoneda: profunctor optics and the Yoneda lemma (functional pearl) | Proceedings of the ACM on Programming Languages

Profunctor optics are a neat and composable representation of bidirectional data accessors, including lenses, and their dual, prisms. The profunctor representation exploits higher-order functions and higher-kinded type constructor classes, but the ...

Profunctor Optics: Modular Data Accessors

CONTEXT: Data accessors allow one to read and write components of a data structure, such as the fields of a record, the variants of a union, or the elements of a container. These data accessors are collectively known as optics; they are fundamental to programs that manipulate complex data. INQUIR...

Seeing double through dependent optics

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...

Generalized Lens Categories via functors $\mathcal{C}^{\rm op}\to\mathsf{Cat}$

Lenses have a rich history and have recently received a great deal of attention from applied category theorists. We generalize the notion of lens by defining a category $\mathsf{Lens}_F$ for any category $\mathcal{C}$ and functor $F\colon \mathcal{C}^{\rm op}\to\mathsf{Cat}$, using a variant of the Grothendieck construction. All of the mathematics in this note is straightforward; the purpose is simply to see lenses in a broader context where some closely-related examples, such as ringed spaces and open continuous dynamical systems, can be included.

Limits and Colimits in a Category of Lenses

Lenses are an important tool in applied category theory. While individual lenses have been widely used in applications, many of the mathematical properties of the corresponding categories of lenses have remained unknown. In this paper, we study the category of small categories and asymmetric delta lenses, and prove that it has several good exactness properties. These properties include the existence of certain limits and colimits, as well as so-called imported limits, such as imported products and imported pullbacks, which have arisen previously in applications. The category is also shown to be extensive, and it has an image factorisation system.

Symmetric lenses | Proceedings of the 38th annual ACM SIGPLAN-SIGACT symposium on Principles of programming languages

A lens is a bidirectional transformation between a pair of connected data structures, capable of translating an edit on one structure into an appropriate edit on the other. Many varieties of lenses have been studied, but none, to date, has offered a ...

Quotient lenses | Proceedings of the 13th ACM SIGPLAN international conference on Functional programming

Bidirectional programming languages are a practical approach to the view update problem. Programs in these languages, called lenses, define both a view and an update policy - i.e., every program can be read as a function mapping sources to views as well ...

Delta Lenses over Inductive Types

Existing bidirectional languages are either state-based or operation-based, depending on whether they represent updates as mere states or as sequences of edit operations.In-between both worlds are delta-based frameworks, where updates are represented using alignment relationships between states.In this paper, we formalize delta lenses over inductive types using dependent type theory and develop a point-free delta lens language with an explicit separation of shape and data.In contrast with the already known issue of data alignment, we identify the new problem of shape alignment and solve it by lifting standard recursion patterns such as folds and unfolds to delta lenses that use alignment to infer meaningful shape updates.

Bidirectional Programming Languages

The need to edit source data through a view arises in a host of applications across many different areas of computing. Unfortunately, few existing systems provide support for updatable views. In practice, when they are needed, updatable views are usually implemented using two separate programs: one that computes the view from the source and another that handles updates. This rudimentary design is tedious for programmers, difficult to reason about, and a nightmare to maintain. This dissertation presents bidirectional programming languages, which provide an elegant and effective mechanism for describing updatable views. Unlike programs written in an ordinary language, which only work in one direction, programs in a bidirectional language can be run both forwards and backwards: from left to right, they describe functions that map sources to views, and from right to left, they describe functions that map updated views back to updated sources. Besides eliminating redundancy, these languages can be designed to ensure correctness, guaranteeing by construction that the two functions work well together. Starting from the foundations, we define a general semantic space of well-behaved bidirectional transformations called lenses. Then, building on this foundation, we describe a particular language for defining lenses on strings with syntax based on the familiar regular operators (union, concatenation, and Kleene star). We present extensions to the basic framework that address the subtle complications that arise when lenses are used to manipulate, data containing unimportant details, ordered data, and confidential data.

Project Cambria: Translate your data with lenses

Changing schemas in distributed software is hard. Could adopting bidirectional lenses help?

Functorial data migration

In this paper we present a simple database definition language: that of categories and functors. A database schema is a small category and an instance is a set-valued functor on it. We show that morphisms of schemas induce three “data migration functors”, which translate instances from one schema to the other in canonical ways. These functors parameterize projections, unions, and joins over all tables simultaneously and can be used in place of conjunctive and disjunctive queries. We also show how to connect a database and a functional programming language by introducing a functorial connection between the schema and the category of types for that language. We begin the paper with a multitude of examples to motivate the definitions, and near the end we provide a dictionary whereby one can translate database concepts into category-theoretic concepts and vice versa.

Categories for the Working Mathematician

Categories for the Working Mathematician provides an array of general ideas useful in a wide variety of fields. Starting from the foundations, this book illuminates the concepts of category, functor, natural transformation, and duality. The book then turns to adjoint functors, which provide a description of universal constructions, an analysis of the representations of functors by sets of morphisms, and a means of manipulating direct and inverse limits. These categorical concepts are extensively illustrated in the remaining chapters, which include many applications of the basic existence theorem for adjoint functors. The categories of algebraic systems are constructed from certain adjoint-like data and characterized by Beck's theorem. After considering a variety of applications, the book continues with the construction and exploitation of Kan extensions. This second edition includes a number of revisions and additions, including two new chapters on topics of active interest. One is onsymmetric monoidal categories and braided monoidal categories and the coherence theorems for them. The second describes 2-categories and the higher dimensional categories which have recently come into prominence. The bibliography has also been expanded to cover some of the many other recent advances concerning categories.

Categorical Data Structures for Technical Computing

Many mathematical objects can be represented as functors from finitely-presented categories $\mathsf{C}$ to $\mathsf{Set}$. For instance, graphs are functors to $\mathsf{Set}$ from the category with two parallel arrows. Such functors are known informally as $\mathsf{C}$-sets. In this paper, we describe and implement an extension of $\mathsf{C}$-sets having data attributes with fixed types, such as graphs with labeled vertices or real-valued edge weights. We call such structures "acsets," short for "attributed $\mathsf{C}$-sets." Derived from previous work on algebraic databases, acsets are a joint generalization of graphs and data frames. They also encompass more elaborate graph-like objects such as wiring diagrams and Petri nets with rate constants. We develop the mathematical theory of acsets and then describe a generic implementation in the Julia programming language, which uses advanced language features to achieve performance comparable with specialized data structures.