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SUMMARY:Hypercovers and factorisation systems in higher categories - Alexa
 nder Campbell (Macquarie University)
DTSTART:20150825T131500Z
DTEND:20150825T141500Z
UID:TALK60468@talks.cam.ac.uk
CONTACT:Zhen Lin Low
DESCRIPTION:Grothendieck's plus construction sends a presheaf F to the pre
 sheaf of equivalence classes of "matching families" of elements of F\; two
  iterations yield the associated sheaf of F (n + 2 iterations are generall
 y required for n-stacks). This construction involves colimits over covers.
  If we instead take colimits over hypercovers\, only one iteration is requ
 ired (for all n)\; this corresponds to taking equivalence classes of "loca
 lly matching famillies" of elements. A similar construction appears in Ver
 dier's hypercovering theorem on abelian sheaf cohomology.\n\nIn this talk 
 I will give an account of hypercovers within the higher categorical theory
  of non-abelian cohomology introduced in my previous talk. I will define (
 n-)hypercovers over a site to be the indexed n-functors which are locally 
 k-surjective for all k in { 0\, 1\, ...\, n }. These maps form the left cl
 ass of a factorisation system which will allow us to see the plus construc
 tion over hypercovers as another generalised Lawvere construction\; for th
 e presheaf case the relevant hypercovers are the locally eso and locally f
 ull functors. I will also compare this approach to the simplicial theory o
 f hypercovers.
LOCATION:MR5\, Centre for Mathematical Sciences
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