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CATEGORIES:Talks on Category Theory
SUMMARY:The Fundamental Theorem and Cauchy Completeness -
Filip Bár (University of Cambridge)
DTSTART;TZID=Europe/London:20121126T170000
DTEND;TZID=Europe/London:20121126T180000
UID:TALK41757AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/41757
DESCRIPTION:We will formulate and proof the Fundamental Theore
m of Category Theory. This theorem (and related th
eorems) lie at the heart of many applications of C
ategory Theory to other fields of mathematics. It
is\nalso an important technical tool in various su
bfields of Category Theory itself. We shall presen
t some of the examples and point out where we have
encountered the Fundamental Theorem in secret on
the example sheets already.\n\nThe fundamental the
orem stresses once more the importance of functor
categories of the form [C^op^\, Set] for a small c
ategory C and the accompanying Yoneda embedding. W
e can hence ask the question on\nnecessary and suf
ficient conditions for a category E to be equivale
nt to a functor category of this form. In particul
ar\, we need to ask when we can recover C from [C^
op^\,Set]. It turns out that C can be recovered if
f it is Cauchy complete. In the second part of thi
s talk we shall present various equivalent descrip
tions of the Cauchy completion of a category.\n\nI
f there is time\, we shall consider metric spaces
as (enriched) categories and sketch why the Cauchy
completion of a metric space considered as a cate
gory is the familiar Cauchy completion of a metric
spaces to a complete metric space as encountered
in Analysis\, Functional Analysis and Topology.
LOCATION:MR12
CONTACT:Filip Bár
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