Abstract: One of the first applications of model categories was Quillen homology. Building on the notion of Beck modules, one defines the cotangent complex of an associative or commutative (dg)-algebras as the derived functor of its abelianization. The latter is a module over the original algebra, and its homology groups are called the (Andre'-)Quillen homology. The caveat of this approach is that the cotangent complex is not defined as a functor on the category of all algebras. To remedy this, Lurie's "cotangent complex formalism" (Higher Algebra & 7) uses the 00-categorical Grothendieck construction and gives a general treatment for the cotangent complex of an algebra over a (coherent) 00-operad.
In this talk I will propose a way to parallel Lurie's approach using model categories which is based on the model-categorical Grothendieck construction as developed by Yonatan Harpaz and myself. In particular, we will see that the cotangent complex of an algebra over a (dg)-operad, may be defined as the total derived functor of a left Quillen functor. At the cost of restricting the generality, our approach offers a simplification to that of Lurie in that one can avoid carrying a significant amount of coherent data.
I will assume basic familiarity with model categories but not much more.
This is a joint work with Yonatan Harpaz and Joost Nuiten.