Operads, Grothendieck topologies and Deformation theory

TL;DR

This paper introduces a sheaf-based cohomology framework on a specific site to unify deformation theory and cohomology without explicit cocycles, enabling new proofs of the PBW theorem.

Contribution

It develops an invariant cohomology approach using sheaves on a site, allowing simultaneous treatment of various algebra types and proving the PBW theorem in new cases.

Findings

Unified cohomology theory for deformation and algebraic structures

Proved PBW theorem in previously unknown cases

Provides a cocycle-free method for deformation cohomology

Abstract

The idea of the work is to find an invariant way to pass from deformation theory to cohomology, which does not use any explicit cocycles. The appropriate cohomology theory is based on considering sheaves on a certain site. An advantage of the approach is that it enables one to give a simultanious treatement to all types of algebras. As an application, we prove the PBW theorem in cases where it is not yet known.