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

Series: 
Geometry Topology Seminar
Monday, December 8, 2014 - 14:00
1 hour (actually 50 minutes)
Location: 
Skiles 006
,  
Harvard University
Organizer: 
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.