The more legs the merrier: A new composition for symmetric (multi-)lenses

Johnson, Michael; Rosebrugh, Robert

doi:10.4204/eptcs.333.7

Electron. Proc. Theor. Comput. Sci.

2021

DOI: 10.4204/eptcs.333.7

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The more legs the merrier: A new composition for symmetric (multi-)lenses

Michael Johnson

Robert Rosebrugh

Abstract: This paper develops a new composition of symmetric lenses that preserves information which is important for implementing system interoperation. It includes a cut-down but realistic example of a multi-system business supply chain and illustrates the new mathematical content with analysis of the systems, showing how the new composition facilitates the engineering required to implement the interoperations. All of the concepts presented here are based on either pure category theory or on experience in solving busi… Show more

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Cited by 2 publications

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Coequalisers under the Lens

Meglio

2022

Electron. Proc. Theor. Comput. Sci.

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Lenses encode protocols for synchronising systems. We continue the work begun by Chollet et al. at the Applied Category Theory Adjoint School in 2020 to study the properties of the category of small categories and asymmetric delta lenses. The forgetful functor from the category of lenses to the category of functors is already known to reflect monos and epis and preserve epis; we show that it preserves monos, and give a simpler proof that it preserves epis. Together this gives a complete characterisation of the monic and epic lenses in terms of elementary properties of their get functors.Next, we initiate the study of coequalisers of lenses. We observe that not all parallel pairs of lenses have coequalisers, and that the forgetful functor from the category of lenses to the category of functors neither preserves nor reflects all coequalisers. However, some coequalisers are reflected; we study when this occurs, and then use what we learned to show that every epic lens is regular, and that discrete opfibrations have pushouts along monic lenses. Corollaries include that every monic lens is effective, every monic epic lens is an isomorphism, and the class of all epic lenses and the class of all monic lenses form an orthogonal factorisation system.

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Coequalisers under the Lens

Meglio

2022

Electron. Proc. Theor. Comput. Sci.

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Limits and Colimits in a Category of Lenses

Chollet¹,

Clarke²,

Johnson³

et al. 2022

Electron. Proc. Theor. Comput. Sci.

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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.

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The more legs the merrier: A new composition for symmetric (multi-)lenses

Cited by 2 publications

References 15 publications

Coequalisers under the Lens

Coequalisers under the Lens

Limits and Colimits in a Category of Lenses

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