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DTSTAMP:20250723T104604Z
DESCRIPTION:Title: Some remarks on simplicial sets and certain related cate
 gories\n\nAbstract: In my three-part paper 'Generalized sketches as a fram
 ework for completeness theorems' (JPAA 1997)\, I construct\, for each of a
  number of categorical doctrines\, call it D\, a presheaf category C such 
 that D is the full subcategory of C with objects that are injective relati
 ve to a small\, usually finite\, number of arrows\, the 'sketch-axioms'\, 
 in C \; the set of the sketch-axioms I denote by A . For example\, if D is
  the category of small finite-limit categories (with arrows the functors p
 reserving finite limits in the non-strict sense)\, than C is the category 
 (with suitable arrows!) of finite-limit sketches. In each of the examples 
 of D \, one has two weak factorization systems (the factoring diagonal is 
 not required to be unique)\, one of them giving rise\, using the above set
  A of 'sketch-axioms' to the objects of D as the Kan complexes arise as th
 e fibrant objects\, from the horn-extensions in the Quillen model structur
 e on simplicial sets. I am interested in the question for which of my exam
 ples of sketch-categories C the two factorization systems determine a Quil
 len model structure\; in some simple cases\, I already know that this is c
 ase. In the sketch-categories\, the strict anodyne maps play a distinguish
 ed role. These are the ones that\, in the classical case of simplicial set
 s\, are obtained from the Gabriel-Zisman definition of anodyne map by omit
 ting reference to retracts. In the sketch-category C\, the strict anodyne 
 maps are the transfinite composites of pushouts of the sketch-axioms\, the
  arrows in the set A . (In the classical case also\, the strict anodynes a
 re the transfinite composites of the pushouts of the horn-extensions.) In 
 the talk\, I will start with discussing the classical case of the category
  of simplicial sets with a special emphasis on the strict anodyne maps\, a
 nd a variant of the latter related to Andre Joyal's model structure on sim
 plicial sets where the fibrant object are the quasi-categories.\n
DTSTART:20191008T183000Z
DTEND:20191008T193000Z
LOCATION:Room 920\, Burnside Hall\, CA\, QC\, Montreal\, H3A 0B9\, 805 rue 
 Sherbrooke Ouest
SUMMARY:Michael Makkai (9IÖÆ×÷³§Ãâ·Ñ)
URL:/mathstat/channels/event/michael-makkai-mcgill-301
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