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SUMMARY:The Fundamental Theorem and Cauchy Completeness - Filip Bár (Univ
 ersity of Cambridge)
DTSTART:20121126T170000Z
DTEND:20121126T180000Z
UID:TALK41757@talks.cam.ac.uk
CONTACT:Filip Bár
DESCRIPTION:We will formulate and proof the Fundamental Theorem of Categor
 y Theory. This theorem (and related theorems) lie at the heart of many app
 lications of Category Theory to other fields of mathematics. It is\nalso a
 n important technical tool in various subfields of Category Theory itself.
  We shall present some of the examples and point out where we have encount
 ered the Fundamental Theorem in secret on the example sheets already.\n\nT
 he fundamental theorem stresses once more the importance of functor catego
 ries of the form [C^op^\, Set] for a small category C and the accompanying
  Yoneda embedding. We can hence ask the question on\nnecessary and suffici
 ent conditions for a category E to be equivalent to a functor category of 
 this form. In particular\, we need to ask when we can recover C from [C^op
 ^\,Set]. It turns out that C can be recovered iff it is Cauchy complete. I
 n the second part of this talk we shall present various equivalent descrip
 tions of the Cauchy completion of a category.\n\nIf there is time\, we sha
 ll consider metric spaces as (enriched) categories and sketch why the Cauc
 hy completion of a metric space considered as a category is the familiar C
 auchy completion of a metric spaces to a complete metric space as encounte
 red in Analysis\, Functional Analysis and Topology.
LOCATION:MR12
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