23 June 2017; 12:00; Seminar room C3b/158.
David White. Model Categories and the Grothendieck Construction.Abstract: I will report on joint work with Michael Batanin studying the homotopy theory of the Grothendieck construction, given a category B and a functor F from B to CAT. From the Grothendieck construction we produce a “horizontal” model structure on the base B and “vertical” model structures on the fibers F(b). I will focus on examples, including pairs (R,A) where R is a (commutative) monoid and A is an R-module, pairs (P,A) where P is a (symmetric or non-symmetric) colored operad and A is a P-algebra, and pairs (T,A) where T is a 2-monad on Cat with rank and A is a T-algebra. I will also discuss how to get a semi-model structure under extremely general conditions. Additionally, we study when these model structures are left proper, and when a weak equivalence in B gives rise to a Quillen equivalence of fibers. Applications include change of rings, rectification of operad-algebras, strictification for categorical structures, and preservation of algebraic structure under left Bousfield localization. I will also explain the relationship of this work to that of Harpaz and Prasma.