Director of Research (if dissertation) or Advisor (if thesis)
Rezk, Charles
Doctoral Committee Chair(s)
Ando, Matthew
Committee Member(s)
McCarthy, Randy
Berwick-Evans, Dan
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Homotopy Theory
Higher Category Theory
Topos Theory
Abstract
The end goal of this work is to define and study an elementary higher topos. We will achieve this by going through several steps. First we review complete Segal spaces. Then we study various fibrations of complete Segal spaces and use that to define representable Cartesian fibrations. Next we use representable Cartesian fibrations to define complete Segal objects which are are model for internal higher categories.
Having done all this work we can then define an elementary higher topos which simultaneously generalizes an elementary topos and higher topos. Then we use all the tools we previously developed to show it satisfies classical topos theoretic properties, such being locally Cartesian closed and descent. Finally we show we can classify univalent maps in an elementary higher topos.
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