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SUMMARY:Generalising the functor of points approach - Zhen Lin Low - DPMMS
  
DTSTART:20150519T131500Z
DTEND:20150519T141500Z
UID:TALK59543@talks.cam.ac.uk
CONTACT:Dr Ignacio Lopez Franco
DESCRIPTION:The passage from commutative rings to schemes has three main s
 teps: first\,\none identifies a distinguished class of ring homomorphisms 
 corresponding to\nopen immersions of schemes\; second\, one defines the no
 tion of an open\ncovering in terms of these distinguished homomorphisms\; 
 and finally\, one\nembeds the opposite of the category of commutative ring
 s in an ambient\ncategory in which one can glue (the formal duals of) comm
 utative rings\nalong (the formal duals of) distinguished homomorphisms. Tr
 aditionally\, the\nambient category is taken to be the category of locally
  ringed spaces\, but\nfollowing [Demazure and Gabriel]\, one could equally
  well work in the\ncategory of sheaves for the large Zariski site – this
  is the so-called\n'functor of points approach'.\n\nThe three procedures d
 escribed above can be generalised to other contexts.\nThe first step essen
 tially amounts to reconstructing the class of open\nembeddings from the cl
 ass of closed embeddings. Once we have a suitable\nclass of open embedding
 s\, the class of open coverings is a subcanonical\nGrothendieck pretopolog
 y. We then define a notion of 'charted space' in the\ncategory of sheaves.
  This gives a uniform way of defining locally Hausdorff\nspaces\, schemes\
 , locally finitely presented C^\\infty-schemes etc. as\nspecial sheaves on
  their respective categories of local models\, taking as\ninput just the c
 lass of closed embeddings. We can also get many variations\non manifolds b
 y skipping the first step and working directly with a given\nclass of open
  embeddings.
LOCATION:MR5\, Centre for Mathematical Sciences
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