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SUMMARY:David Spivak (Topos Institute)
DTSTART:20240802T153000Z
DTEND:20240802T163000Z
DTSTAMP:20260422T171807Z
UID:CompMath/4
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/CompMath/4/"
 >How the Yoneda lemma applies</a>\nby David Spivak (Topos Institute) as pa
 rt of Relatorium seminar\n\n\nAbstract\nWhat is the relationship between y
 our web of concepts about the world and all the examples you've seen of th
 ese concepts? And what is the relationship between a generic flower and al
 l the particular flowers or a generic bicycle and all the particular bicyc
 les? 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.
 \n\nIn this talk\, I'll begin by explaining schemas and instances—concep
 t-webs and the system of examples that live in them—in terms of categori
 es 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. \n\nSo given a concept\, how is the generic i
 nstance 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 particul
 ar 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 (s
 ystem 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 wa
 ys that the generic instance for c can be overlaid onto the system of exam
 ples. And this is how the Yoneda lemma "applies"!\n
LOCATION:https://researchseminars.org/talk/CompMath/4/
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