# QUILLEN COHOMOLOGY OF ENRICHED OPERADS

Abstract : A modern insight due to Quillen, which is further developed by Lurie, asserts that many cohomology theories of interest are particular cases of a single construction, which allows one to define cohomology groups in an abstract setting using only intrinsic properties of the category (or infinity category) at hand. This universal cohomology theory is known as Quillen cohomology. In any setting, Quillen cohomology of a given object is classified by its cotangent complex. The main purpose of this document is to study Quillen cohomology of enriched operads, when working in the model categorical framework. Our main result provides an explicit formula for computing Quillen cohomology of enriched operads, based on a procedure of taking certain infinitesimal models of their cotangent complexes. We are particularly interested in the Quillen cohomology of simplicial operads and dg operads. There is a natural construction of twisted arrow infinity category of a simplicial operad, which extends the notion of twisted arrow infinity category of an infinity category introduced by Lurie. We assert that the cotangent complex of a simplicial operad can be represented as a spectrum valued functor on its twisted arrow infinity category. Turning to the context of dg operads, the situation becomes simpler due to the stability of dg modules. We find that the cotangent complex of a dg operad P can be represented by a nice infinitesimal P-bimodule, which is in fact closely related to the module of Kähler differentials of P via a cofiber sequence. Moreover, we prove the existence of an operadic version of the Dold-Kan correspondence, then due to this we find a connection between Quillen cohomology of a simplicial operad and Quillen cohomology of its associated dg operad. In the last section, we establish the relation between deformation theory and Quillen cohomology.
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https://hal.archives-ouvertes.fr/tel-03311677
Contributor : Truong Hoang <>
Submitted on : Wednesday, August 4, 2021 - 12:26:26 AM
Last modification on : Friday, August 6, 2021 - 3:22:22 AM

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Thesis-HoangTruong.final.pdf
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• HAL Id : tel-03311677, version 1

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Truong Hoang. QUILLEN COHOMOLOGY OF ENRICHED OPERADS. Algebraic Topology [math.AT]. Université Sorbonne Paris Nord, 2021. English. ⟨tel-03311677⟩

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