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UID:4b0e414bc3572c39ed4a7466d4a673f1
CATEGORIES:Colloquia
CREATED:20230926T102316
SUMMARY:Do we need a new foundation for higher structures?
LOCATION:Hill 705
DESCRIPTION:Emily Riehl (Johns Hopkins U.)\nTitle: Do we need a new foundation for high
 er structures?\nAbstract: The fundamental theorem of category theory is the
  Yoneda lemma, which in its simplest form identifies natural transformation
 s between represented functors with morphisms between the representing obje
 cts. The ?-categorical Yoneda lemma is surprisingly hard to prove --- at le
 ast in the traditional set-based foundations of mathematics. In this talk w
 e'll describe the experience of developing ?-category theory in an alternat
 e foundation system based on homotopy type theory, in which constructions d
 etermined up to a contractible space of choices are genuinely "well-defined
 " and elementwise mappings are automatically homotopically-coherently funct
 orial. In this setting, the proof the ?-categorical Yoneda lemma is arguabl
 y easier than the 1-categorical Yoneda lemma. We'll end by posing the quest
 ion as to whether similar foundations would be useful for other "higher str
 uctures." This is based on joint work with Mike Shulman and involves comput
 er formalizations written in collaboration with Nikolai Kudasov and Jonatha
 n Weinberger.\n \n
X-ALT-DESC;FMTTYPE=text/html:<p>Emily Riehl (Johns Hopkins U.)</p><p>Title: Do we need a new foundation 
 for higher structures?</p><p>Abstract: The fundamental theorem of category 
 theory is the Yoneda lemma, which in its simplest form identifies natural t
 ransformations between represented functors with morphisms between the repr
 esenting objects. The ?-categorical Yoneda lemma is surprisingly hard to pr
 ove --- at least in the traditional set-based foundations of mathematics. I
 n this talk we'll describe the experience of developing ?-category theory i
 n an alternate foundation system based on homotopy type theory, in which co
 nstructions determined up to a contractible space of choices are genuinely 
 "well-defined" and elementwise mappings are automatically homotopically-coh
 erently functorial. In this setting, the proof the ?-categorical Yoneda lem
 ma is arguably easier than the 1-categorical Yoneda lemma. We'll end by pos
 ing the question as to whether similar foundations would be useful for othe
 r "higher structures." This is based on joint work with Mike Shulman and in
 volves computer formalizations written in collaboration with Nikolai Kudaso
 v and Jonathan Weinberger.</p><p>&nbsp;</p>
CONTACT:Emily Riehl
DTSTAMP:20260830T172138
DTSTART;TZID=America/New_York:20231004T153000
DTEND;TZID=America/New_York:20231004T163000
SEQUENCE:0
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