Simplicial sets in perspective

Andre Joyal
Monday, July 22, 2013 - from 09:30 to 10:30
The notion of simplicial set was introduced by Eilenberg and Zilber in 1950, originally for applications to algebraic topology. It was a giant step forward, since the notion is currently used in many branches of mathematics, from homotopical algebra to algebraic geometry, higher category theory and constructive logic. For instance, it was used by Daniel Kan for the foundation of combinatorial homotopy theory, by Daniel Quillen for homotopical algebra, by Bertrand Toën and Gabriele Vezzosi for homotopical algebraic geometry, by Jacob Lurie for higher topos theory and by Vladimir Voevodsky for the semantics of Martin-Löf type theory. The foundation of higher category theory is currently based on simplicial structures of different forms: simplicial categories, Segal categories, Rezk categories and quasi-categories. The latter is just a simplicial set satisfying the Boardman condition. After briefly recalling the history of simplicial sets, I will describe their applications to higher category theory and then mention some recent generalizations.