How the Yoneda lemma applies

Abstract: What is the relationship between your web of concepts about the world and all the examples you've seen of these concepts? And what is the relationship between a generic flower and all the particular flowers or a generic bicycle and all the particular bicycles? A formal answer to this was given by Nobuo Yoneda in a private letter to a founder of Category Theory, Saunders Mac Lane, and this answer has become the most fundamental concept in category theory: the Yoneda lemma.

In this talk, I'll begin by explaining schemas and instances—concept-webs and the system of examples that live in them—in terms of categories C and set-valued functors F:C-->Set. Then I'll explain how each concept (each node in the web) determines a generic instance: the generic flower, the generic bicycle, etc.

So given a concept, how is the generic instance of it related to all the other examples of it? The answer is that the generic instance of flower can be overlaid perfectly onto any particular flower, and all its generic features will be given particular values. This is the content of the Yoneda lemma: given any schema C and functor (system of examples) F: C-->Set, the Yoneda lemma says that "applying F to concept c", i.e. the set of c-examples, is the same as the set of all ways that the generic instance for c can be overlaid onto the system of examples. And this is how the Yoneda lemma "applies"!

Computer scienceMathematics

Audience: researchers in the topic

( video )

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Relatorium seminar

Series comments: The name "Relatorium" combines "relator" with the Latin root "-ium," meaning "a place for activities" (as in "auditorium" or "gymnasium"). This seminar series is a platform to relate ideas, interact with math, and connect with each other.

In this series, we explore math beyond what we usually hear in standard talks. These sessions fall somewhere between a technical talk and a podcast: moderately formal, yet conversational. The philosophy behind the series is that math is best learned by active participation rather than passive listening. Our aim is to “engage and involve,” inviting everyone to think actively with the speaker. The concepts are accessible, exploratory, and intended to spark questions and discussions.

The idea of relatability has strong ties to compassion — creating space for shared understanding and exploration - which is the spirit of this seminar! This is a pilot project, so we’re here to improvise, learn, and evolve as we go!

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