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YouTube

Defects in Category Theory: From Functors to Proarrow Equipments

Topos Institute via YouTube

Overview

Watch a Berkeley seminar presentation exploring the mathematical concept of defects in functor theory, from their introduction by Auslander in the 1960s to modern developments. Delve into how defects can be defined for enriched functors with cosmos codomain, examining both covariant and contravariant cases. Learn about the relationship between defect functors and Yoneda embeddings through adjoint properties, and discover how these concepts extend to profunctors enriched in cosmos V. Explore the connection between Isbell duals and defects, understanding their determination through covariant and contravariant properties. Follow the application of Tensor-Hom-Cotensor adjunctions and (co)end calculus, culminating in the generalization of defects to proarrow equipments. The 58-minute talk systematically covers fundamental concepts, including profunctor definitions, composition, bitensored categories, and the Yoneda and coYoneda lemmas, providing a comprehensive exploration of this advanced mathematical topic.

Syllabus

Intro
The defect of a covariant f.p. functor
The defect of a contravariant f.p. functor
The defect of a covariant additive functor
The defect of a contravariant additive functor
Questions
Outline of the results
Conventions
Main tools
Definition of profunctor
Examples of profunctors
Composition of profunctors
Triangle of profunctors
Bitensored category
The Yoneda and coYoneda lemmas
Defects of a profunctor
Adjunctions for defects
The contravariant defect of a f.p. functor (revisited)
The contravariant defect of an additive functor (revisited)
The covariant defect of an additive functor (revisited)
Isbell duality interchanges representable functors
Isbell duals for profunctors
Isbell duals factor through defects
Equipment
Defects are adjoint to Yoneda embeddings

Taught by

Topos Institute

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