Matan Prasma: A model-categorical cotangent complex formalism

ABSTRACT

One of the first applications of the theory 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 potentially restricting 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.