Profunctor Optics: Modular Data Accessors

Dependent Optics

A wide variety of bidirectional data accessors, ranging from mixed optics to functor lenses, can be formalized within a unique framework-dependent optics. Starting from two indexed categories, which encode what maps are allowed in the forward and backward directions, we define the category of dependent optics and establish under what assumptions it has coproducts. Different choices of indexed categories correspond to different families of optics: we discuss dependent lenses and prisms, as well as closed dependent optics. We introduce the notion of Tambara representation and use it to classify contravariant functors from the category of optics, thus generalizing the profunctor encoding of optics to the dependent case.

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 ...

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.

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 ...

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...

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 ...

Project Cambria: Translate your data with lenses

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

Relational lenses | Proceedings of the twenty-fifth ACM SIGMOD-SIGACT-SIGART symposium on Principles of database systems

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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.

(PDF) The Comparative Anatomy of Nanopublications and FAIR Digital Objects

PDF | Beginning in 1995, early Internet pioneers proposed Digital Objects as encapsulations of data and metadata made accessible through persistent... | Find, read and cite all the research you need on ResearchGate

Solid Protocol

This document connects a set of specifications that, together, provide applications with secure and permissioned access to externally stored data in an interoperable way.

Capability Trees: A Protocol-Level Extension of Object Capabilities, Draft 3

This draft incorporates feedback from Brooklyn Zelenka (@expede.wtf), Daniel Holmgren (@dholms.at), and Zicklag (@zicklag.dev), and addresses questions raised in the ATProto Private Data Working Group. Draft 2 remains available for transparency.

opticdev/optic

OpenAPI linting, diffing and testing. Optic helps prevent breaking changes, publish accurate documentation and improve the design of your APIs.

Concise Binary Object Representation (CBOR)

The Concise Binary Object Representation (CBOR) is a data format whose design goals include the possibility of extremely small code size, fairly small message size, and extensibility without the need for version negotiation. These design goals make it different from earlier binary serializations such as ASN.1 and MessagePack. This document obsoletes RFC 7049, providing editorial improvements, new details, and errata fixes while keeping full compatibility with the interchange format of RFC 7049. It does not create a new version of the format.