BEGIN:VCALENDAR
VERSION:2.0
PRODID:-//talks.cam.ac.uk//v3//EN
BEGIN:VTIMEZONE
TZID:Europe/London
BEGIN:DAYLIGHT
TZOFFSETFROM:+0000
TZOFFSETTO:+0100
TZNAME:BST
DTSTART:19700329T010000
RRULE:FREQ=YEARLY;BYMONTH=3;BYDAY=-1SU
END:DAYLIGHT
BEGIN:STANDARD
TZOFFSETFROM:+0100
TZOFFSETTO:+0000
TZNAME:GMT
DTSTART:19701025T020000
RRULE:FREQ=YEARLY;BYMONTH=10;BYDAY=-1SU
END:STANDARD
END:VTIMEZONE
BEGIN:VEVENT
CATEGORIES:Category Theory Seminar
SUMMARY:Generalising the functor of points approach - Zhen
Lin Low - DPMMS
DTSTART;TZID=Europe/London:20150519T141500
DTEND;TZID=Europe/London:20150519T151500
UID:TALK59543AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/59543
DESCRIPTION:The passage from commutative rings to schemes has
three main steps: first\,\none identifies a distin
guished class of ring homomorphisms corresponding
to\nopen immersions of schemes\; second\, one defi
nes the notion of an open\ncovering in terms of th
ese distinguished homomorphisms\; and finally\, on
e\nembeds the opposite of the category of commutat
ive rings in an ambient\ncategory in which one can
glue (the formal duals of) commutative rings\nalo
ng (the formal duals of) distinguished homomorphis
ms. Traditionally\, the\nambient category is taken
to be the category of locally ringed spaces\, but
\nfollowing [Demazure and Gabriel]\, one could equ
ally well work in the\ncategory of sheaves for the
large Zariski site – this is the so-called\n'func
tor of points approach'.\n\nThe three procedures d
escribed above can be generalised to other context
s.\nThe first step essentially amounts to reconstr
ucting the class of open\nembeddings from the clas
s of closed embeddings. Once we have a suitable\nc
lass of open embeddings\, the class of open coveri
ngs is a subcanonical\nGrothendieck pretopology. W
e then define a notion of 'charted space' in the\n
category of sheaves. This gives a uniform way of d
efining locally Hausdorff\nspaces\, schemes\, loca
lly finitely presented C^\\infty-schemes etc. as\n
special sheaves on their respective categories of
local models\, taking as\ninput just the class of
closed embeddings. We can also get many variations
\non manifolds by skipping the first step and work
ing directly with a given\nclass of open embedding
s.
LOCATION:MR5\, Centre for Mathematical Sciences
CONTACT:Dr Ignacio Lopez Franco
END:VEVENT
END:VCALENDAR