# Double Category Published articles for Double Category. This is one page of public article previews, not the complete archive. Follow Next page to continue. Summaries are not the original full articles. ## Yoneda Lemma in Double Categories DevFeed: [Yoneda Lemma in Double Categories]() Original publisher: [Read original article]() Author: Bartosz Milewski Published: 2026-09-13T12:23:31Z Content type: article Language: en Sources: [Bartosz Milewski's Programming Cafe]() Topics: [Category Theory](), [Profunctors](), [String Diagrams](), [Haskell]() Tags: [category-theory](), [double-category](), [implementation](), [kan-extensions](), [proarrow-equipment](), [profunctor-equipment](), [profunctors](), [string-diagrams](), [yoneda-structure]() ### AI overview The article explains how to formulate the Yoneda lemma in double categories without directly referring to presheaves or hom-sets. It uses profunctors, universal constructions, Kan extensions, tabulations, 2-cells, and string diagrams to describe the Yoneda embedding and its desired properties, including density and full faithfulness. ### Source excerpt Working with double categories can be aptly summarized in a meme: Talk to me about sets without mentioning sets. We don't talk about hom-sets, we talk about horizontal units. Secretly, we are visualizing horizontal arrows as profunctors, and the unit of profunctor composition is a hom-functor. Presheaves are defined as -valued functors, so we immediately [...] ## Tambara Equipment DevFeed: [Tambara Equipment]() Original publisher: [Read original article]() Author: Bartosz Milewski Published: 2026-07-11T08:13:34Z Content type: article Language: en Sources: [Bartosz Milewski's Programming Cafe]() Topics: [modules](), [Haskell](), [Code]() Tags: [category-theory](), [code](), [double-categories](), [double-category](), [haskell](), [modules](), [optics](), [proarrow-equipment](), [profunctors](), [structure](), [tambara-modules](), [tannakian-reconstruction](), [theory](), [transformation]() ### AI overview This article explains Tambara modules through category theory and illustrates the concepts with Haskell code. It discusses their relationship to profunctors, monoidal actions, double categories, proarrow equipment, and Tannakian reconstruction. ### Source excerpt I was originally attracted to category theory when trying to understand Haskell optics. I was puzzled by the van Laarhoven's functor representations and Kmett's use of Tambara modules. By playing Tetris with the Yoneda lemma I was able to make some progress, attacking more and more esoteric topics. With a group of researcher and students [...] ## Kan Extensions in Double Categories DevFeed: [Kan Extensions in Double Categories]() Original publisher: [Read original article]() Author: Bartosz Milewski Published: 2026-06-13T12:27:28Z Content type: article Language: en Sources: [Bartosz Milewski's Programming Cafe]() Topics: [Haskell](), [data type](), [implementation]() Tags: [category-theory](), [data-type](), [double-category](), [function](), [haskell](), [implementation](), [kan-extensions](), [profunctor-equipment](), [profunctors]() ### AI overview This article generalizes right and left Kan extensions from functors to profunctors in double categories and presents corresponding Haskell representations. It explains the associated universal and factorization properties, including their computational interpretation. ### Source excerpt Previously: Kan extensions in Haskell. In a double category that is also a proarrow equipment, we have the ability to bend arrows. In particular, in the definition of the counit of the right Kan extension: we can bend the vertical arrow, replacing it with its horizontal conjoint . In a profunctor equipment, this is just [...]