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SUMMARY:Fibration categories and models for higher categories - Christian 
 Sattler (University of Leeds)
DTSTART:20170307T141500Z
DTEND:20170307T151500Z
UID:TALK71174@talks.cam.ac.uk
CONTACT:Tamara von Glehn
DESCRIPTION:This talk reports on two related pieces of work in progress. H
 omotopy type theory is a language for certain categories of spaces\, but d
 epends only on the structure of a (type-theoretic) fibration category rath
 er than a model category. Fibration categories are easier to construct tha
 n model categories\, and although they do not support the full range of co
 nstructions known from model categories\, often it is easier to see a high
 er-dimensional entity as an object of a fibration category. For example\, 
 simplicial sets form a model category\, but semisimplicial sets only form 
 a type-theoretic fibration category.\n\nThe first part concerns models of 
 higher categories in (marked) presheaves on a category C. A standard examp
 le is Verity's model structure on weak complicial sets\, where C = Delta. 
 When C is direct\, one typically only obtains a fibration category\, as is
  the case for opetopic sets. We propose a route for relating these two mod
 els by constructing an intermediate model in weak semicomplicial sets.\n\n
 In the second part\, we look at models of higher categories based on space
 -valued presheaves where spaces are now assumed to be modelled by just a (
 type-theoretic) fibration category. Recently\, Paolo Capriotti has suggest
 ed complete semi-Segal spaces as a replacement for the technology of compl
 ete Segal spaces not available in this context\, yielding a notion of univ
 alent (omega\,1)-categories (assuming omega-Reedy limits). This only works
  in the presence of completeness: due to the lack of degeneracies\, semi-S
 egal spaces do not correspond to non-univalent (i.e.\, pre) (omega\,1)-cat
 egories. We define a direct category D such that D-Segal spaces\, space-va
 lued presheaves on D with an analogue of the Segal condition\, fill this r
 ole. There is an analogue of the completeness condition\, and complete D-S
 egal spaces correspond to semi-Segal spaces.
LOCATION:MR5\, Centre for Mathematical Sciences
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