Higher enveloping algebras

Abstract

We provide spectral Lie algebras with enveloping algebras over the operad of little $G$-framed $n$-dimensional disks for any choice of dimension $n$ and structure group $G$, and we describe these objects in two complementary ways. The first description is an abstract characterization by a universal mapping property, which witnesses the higher enveloping algebra as the value of a left adjoint in an adjunction. The second, a generalization of the Poincaré-Birkhoff-Witt theorem, provides a concrete formula in terms of Lie algebra homology. Our construction pairs the theories of Koszul duality and Day convolution in order to lift to the world of higher algebra the fundamental combinatorics of Beilinson-Drinfeld's theory of chiral algebras. Like that theory, ours is intimately linked to the geometry of configuration spaces and has the study of these spaces among its applications. We use it here to show that the stable homotopy types of configuration spaces are proper homotopy invariants.

arXiv commentTo appear in Geometry & Topology. May vary slightly from published version

References in corpus (12)

• Factorization homology of topological manifolds
• The tangent complex and Hochschild cohomology of E_n-rings
• Bar constructions for topological operads and the Goodwillie derivatives of the identity
• Dualizing cartesian and cocartesian fibrations
• Betti numbers and stability for configuration spaces via factorization homology
• The homology of the little disks operad
• Betti numbers of configuration spaces of surfaces
• Zero-pointed manifolds
• Finite subset spaces of S^1
• On the homotopy invariance of configuration spaces
• Day convolution for infinity-categories
• The mod 2 homology of free spectral Lie algebras

Cited by in corpus (13)

• Cohomology of generalized configuration spaces
• Stable $\infty$-Operads and the multiplicative Yoneda lemma
• The Lambrechts-Stanley Model of Configuration Spaces
• Vertex Algebras and Costello-Gwilliam Factorization Algebras
• A geometric approach to equivariant factorization homology and nonabelian Poincaré duality
• Curved Koszul duality of algebras over unital versions of binary operads
• Quillen homology of spectral Lie algebras with application to mod $p$ homology of labeled configuration spaces
• The Lubin-Tate Theory of Configuration Spaces: I
• On the Relation between K- and L-Theory of $C^{\ast }$-Algebras
• On the topologies of the exponential
• Higher representation stability for ordered configuration spaces and twisted commutative factorization algebras
• Derived Poincaré-Birkhoff-Witt theorems (with an appendix by Vladimir Dotsenko)
• Configuration spaces of products