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SUMMARY:Infinitesimal models of theories - Filip Bár (DPMMS)
DTSTART:20161018T131500Z
DTEND:20161018T141500Z
UID:TALK68626@talks.cam.ac.uk
CONTACT:Tamara von Glehn
DESCRIPTION:Can we make precise the idea that the geometry of a space is a
 ffine\, euclidean\, or projective at an infinitesimal level?\n\nYes\, we c
 an\, in principle. In fact\, there is a construction for first-order theor
 ies\, which we call their infinitesimalisation. The models of the infinite
 simalisation may be considered as spaces\, which are models of that theory
  at an infinitesimal level.\n\nWhat is considered to be at an infinitesmal
  level for a space is defined by a structure\, which we call infinitesimal
  structure. For a one-sorted first-order theory the construction of infini
 tesimalisation introduces an infinitesimal structure\, replaces every oper
 ation by a partial operation with the domains of definition specified by t
 he infinitesimal structure\, and every relation is required to factor thro
 ugh the infinitesimal structure. The axioms of the theory are adapted acco
 rdingly.\n\nWith these notions at hand we can show that every formal manif
 old in Synthetic Differential Geometry is an infinitesimal model of the al
 gebraic theory of affine combinations\, and that for every Lie group the s
 pace of points\, which are infinitesimal neighbours of the neutral element
 \, yields an infinitesimal model of a group.
LOCATION:MR5\, Centre for Mathematical Sciences
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