On the duality between "free" and "forgetful” constructions

Geometry Topology Seminar
Monday, December 8, 2014 - 2:00pm
1 hour (actually 50 minutes)
Skiles 006
Harvard University
Groups, rings, modules, and compact Hausdorff spaces have underlying sets ("forgetting" structure)  and admit "free" constructions. Moreover, each type of object is completely characterized by the  shadow of this free-forgetful duality cast on the category of sets, and this syntactic encoding  provides formulas for direct and inverse limits. After we describe a typical encounter with  adjunctions, monads, and their algebras, we introduce a new "homotopy coherent" version of this  adjoint duality together with a graphical calculus that is used to define a homotopy coherent  algebra in quite general contexts, such as appear in abstract homotopy theory or derived algebraic  geometry.