GoKawiilCategory theorist extends Yoneda Lemma to double categories without presheaves
A category theory writeup describes how to construct a Yoneda-style embedding within double categories, a setting where presheaves cannot be directly expressed because objects are treated as abstract 0-cells rather than sets. The author sketches how to define a presheaf object and a Yoneda vertical arrow using universal constructions like horizontal arrows and left Kan extensions instead of direct references to sets or presheaves. A follow-up post is promised that will implement these ideas as working code in Haskell.
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The Yoneda Lemma is a foundational tool in category theory, and extending it to double categories broadens the toolkit available for formal reasoning about structures used in programming language theory and functional programming. This kind of work matters to programming language researchers and Haskell developers who rely on category-theoretic abstractions to design type systems and computational models, since a practical implementation could translate into new libraries or techniques for structuring software.
- Double categories avoid direct reference to sets, complicating standard presheaf-based constructions.
- The author reconstructs Yoneda embedding using universal constructions like left Kan extensions instead of presheaves.
- A follow-up article will show a Haskell implementation of these double-category ideas.
Source: bartoszmilewski.com, 2026-09-14
Published there as: “Yoneda Lemma in Double Categories”
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