Speaker: David White (Denison University) Title: Model Categories and the Grothendieck Construction Place: Seminari C3b Date: 23rd June at 12:00. 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 Rmodule, pairs (P,A) where P is a (symmetric or nonsymmetric) colored operad and A is a Palgebra, and pairs (T,A) where T is a 2monad on Cat with rank and A is a Talgebra. I will also discuss how to get a semimodel 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 operadalgebras, 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.
Speaker: Rune Haugseng (Københavns Universitet) Title: Enriched infinityoperads Place: Seminari C3b Date: 23rd June at 14:30. Abstract: I will discuss joint work with Hongyi Chu setting up foundations for a theory of enriched infinityoperads. In cases of interest such as spectral operads and dgoperads this recovers the usual homotopy theory of enriched operads, but is far better behaved  for example, there are natural infinitycategories of algebras over enriched infinityoperads. Time permitting, I will also discuss work in progress describing enriched infinityoperads as monoids in symmetric sequences; this will hopefully be a good setting for understanding Koszul duality beyond characteristic zero.
See the calendar for upcoming events.
