Yoneda Lemma in Double Categories
This article explores the application of the Yoneda Lemma within the framework of double categories, discussing the challenges of defining presheaves without traditional hom-sets. The author outlines a theoretical approach to generalizing categorical constructions for future implementation in Haskell.
Why it matters
It contributes to advanced mathematical theory, specifically in category theory, which underpins formal methods in computer science and functional programming.
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 run into a problem when trying to describe them in a double category. And without presheaves, we can’t talk about the Yoneda lemma — the workhorse of category theory.
Granted, a lot of standard categorical constructions can be generalized to use profunctors in place of presheaves, with immediate generalization to double categorical settings. This can be done with (weighted) limits, Kan extensions, categories of elements (tabulations), and many others. But sometimes you just need to talk about presheaves without mentioning presheaves.
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