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The cotangent complex formalism

Le : 10/05/2016 14h00
Par : Yonatan Harpaz (ENS)
Lieu : I 001
Lien web :
Résumé : Given an object X in a category C, one defines a Beck module over X to be an abelian group object in the category of objects over X. This innocent looking definition ties together many important concepts. When R is a commutative ring we get the notion of an R-module. When R is an associative ring we get the notion of an R-bimodule. When X is an object in a site and C is the associated topos we obtain the notion of a sheaf of abelian groups on X. In many cases Beck modules provide coefficients for various cohomology theories. This can be made precise by working in a higher categorical setting where it is natural to replace the notion of an abelian group object with that of a spectrum object. This leads to the oo-categorical cotangent complex formalism developed by Lurie, which gives a unified framework for notions such as André-Quillen cohomology, Hochschild cohomology and parametrized spectra, together with their classical applications to deformation theory and obstruction theory. In this talk we will describe some current work in progress with Matan Prasma and Joost Nuiten which establishes a model categorical approach to the subject. This approach can be used, in particular, to obtain useful comparison results for the associated categories of modules when C is the category of algebra objects over an operad in a (not necessarily stable) model category. For examples, our approach allows one to compute the category of (spectral) beck modules over simplicial monoids, simplicial categories and simplicial operads.