# Kan extensions

Published articles for Kan extensions.

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## Yoneda Lemma in Double Categories

DevFeed: [Yoneda Lemma in Double Categories](<https://devfeed.tech/articles/yoneda-lemma-in-double-categories-28865.md>)

Original publisher: [Read original article](<https://bartoszmilewski.com/2026/09/13/yoneda-lemma-in-double-categories/>)

Author: Bartosz Milewski

Published: 2026-09-13T12:23:31Z

Content type: article

Language: en

Sources: [Bartosz Milewski's Programming Cafe](<https://devfeed.tech/sources/bartosz-milewski-s-programming-cafe.md>)

Topics: [Category Theory](<https://devfeed.tech/topics/category-theory.md>), [Profunctors](<https://devfeed.tech/topics/profunctors.md>), [String Diagrams](<https://devfeed.tech/topics/string-diagrams.md>), [Haskell](<https://devfeed.tech/topics/haskell.md>)

Tags: [category-theory](<https://devfeed.tech/tags/category-theory.md>), [double-category](<https://devfeed.tech/tags/double-category.md>), [implementation](<https://devfeed.tech/tags/implementation.md>), [kan-extensions](<https://devfeed.tech/tags/kan-extensions.md>), [proarrow-equipment](<https://devfeed.tech/tags/proarrow-equipment.md>), [profunctor-equipment](<https://devfeed.tech/tags/profunctor-equipment.md>), [profunctors](<https://devfeed.tech/tags/profunctors.md>), [string-diagrams](<https://devfeed.tech/tags/string-diagrams.md>), [yoneda-structure](<https://devfeed.tech/tags/yoneda-structure.md>)

### 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 [...]

## Kan Extensions in Double Categories

DevFeed: [Kan Extensions in Double Categories](<https://devfeed.tech/articles/kan-extensions-in-double-categories-28860.md>)

Original publisher: [Read original article](<https://bartoszmilewski.com/2026/06/13/kan-extensions-in-double-categories/>)

Author: Bartosz Milewski

Published: 2026-06-13T12:27:28Z

Content type: article

Language: en

Sources: [Bartosz Milewski's Programming Cafe](<https://devfeed.tech/sources/bartosz-milewski-s-programming-cafe.md>)

Topics: [Haskell](<https://devfeed.tech/topics/haskell.md>), [data type](<https://devfeed.tech/topics/data-type.md>), [implementation](<https://devfeed.tech/topics/implementation.md>)

Tags: [category-theory](<https://devfeed.tech/tags/category-theory.md>), [data-type](<https://devfeed.tech/tags/data-type.md>), [double-category](<https://devfeed.tech/tags/double-category.md>), [function](<https://devfeed.tech/tags/function.md>), [haskell](<https://devfeed.tech/tags/haskell.md>), [implementation](<https://devfeed.tech/tags/implementation.md>), [kan-extensions](<https://devfeed.tech/tags/kan-extensions.md>), [profunctor-equipment](<https://devfeed.tech/tags/profunctor-equipment.md>), [profunctors](<https://devfeed.tech/tags/profunctors.md>)

### 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 [...]

## Kan Extensions in Haskell

DevFeed: [Kan Extensions in Haskell](<https://devfeed.tech/articles/kan-extensions-in-haskell-28859.md>)

Original publisher: [Read original article](<https://bartoszmilewski.com/2026/06/08/kan-extensions-in-haskell/>)

Author: Bartosz Milewski

Published: 2026-06-08T12:31:58Z

Content type: tutorial

Language: en

Sources: [Bartosz Milewski's Programming Cafe](<https://devfeed.tech/sources/bartosz-milewski-s-programming-cafe.md>)

Topics: [Haskell](<https://devfeed.tech/topics/haskell.md>), [Reflex FRP](<https://devfeed.tech/topics/reflex-frp.md>)

Tags: [building](<https://devfeed.tech/tags/building.md>), [category-theory](<https://devfeed.tech/tags/category-theory.md>), [extensions](<https://devfeed.tech/tags/extensions.md>), [haskell](<https://devfeed.tech/tags/haskell.md>), [kan-extensions](<https://devfeed.tech/tags/kan-extensions.md>), [programming](<https://devfeed.tech/tags/programming.md>), [types](<https://devfeed.tech/tags/types.md>), [universal-arrows](<https://devfeed.tech/tags/universal-arrows.md>)

### AI overview

This article explains Kan extensions in category theory and translates their definitions and constructions into Haskell. It focuses on right Kan extensions, adjunctions, universal arrows, and pointwise constructions.

### Source excerpt

Previously: Tabulation Tribulations. If you think of functor composition as a form of multiplication, Kan extensions are an attempt to construct inverses of this multiplication. But unlike multiplication, composition is not symmetric, so we have extensions that attempt to undo precomposition, and lifts that do the same for postcomposition. Furthermore, there rarely is a single [...]