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DESCRIPTION:Strong conceptual completeness for $aleph$_0-categorical theori
 es.\n\nSuppose we have some process to attach to every model of a first-or
 der theory some (permutation) representation of its automorphism group\, c
 ompatible with elementary embeddings. How can we tell if this is 'definabl
 e'\, i.e. really just the points in all models of some imaginary sort of o
 ur theory? In the '80s\, Michael Makkai provided the following answer to t
 his question: a functor Mod(T) $ o$ Set is definable if and only if it pre
 serves all ultraproducts and all 'formal comparison maps' between them (ge
 neralizing e.g. the diagonal embedding into an ultrapower). This is known 
 as strong conceptual completeness\; formally\, the statement is that the c
 ategory Def(T) of definable sets can be reconstructed up to bi-interpretab
 ility as the category of 'ultrafunctors' Mod(T) $ o$ Set. Now\, any genera
 l framework which reconstructs theories from their categories of models sh
 ould be considerably simplified for $aleph$0-categorical theories. Indeed\
 , we show: If T is $aleph$_0-categorical\, then X : Mod(T) $ o$ Set is def
 inable\, i.e. isomorphic to (M $mapsto$ $phi$(M)) for some formula $phi$ $
 in$ T\, if and only if X preserves ultraproducts and diagonal embeddings i
 nto ultrapowers. This means that all the preservation requirements for ult
 ramorphisms\, which a priori get unboundedly complicated\, collapse to jus
 t diagonal embeddings when T is $aleph$_0-categorical. We show this defina
 bility criterion fails if we remove the $aleph$_0-categoricity assumption\
 , by constructing examples of theories and non-definable functors Mod(T) $
  o$ Set which exhibit this. Time permitting\, I will discuss what ev_A : M
 od(T) $ o$ Set being a (pre)ultrafunctor allows us to deduce about an arbi
 trary object A of the classifying topos E(T).\n
DTSTART:20171205T193000Z
DTEND:20171205T203000Z
LOCATION:Room 920\, Burnside Hall\, CA\, QC\, Montreal\, H3A 0B9\, 805 rue 
 Sherbrooke Ouest
SUMMARY:Jesse Han\, McMaster
URL:/mathstat/channels/event/jesse-han-mcmaster-283162
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