Unifying derived deformation theories

Abstract

We develop a framework for derived deformation theory, valid in all characteristics. This gives a model category reconciling local and global approaches to derived moduli theory. In characteristic 0, we use this to show that the homotopy categories of DGLAs and SHLAs (L ∞ -algebras) considered by Kontsevich, Hinich and Manetti are equivalent, and are compatible with the derived stacks of Toën-Vezzosi and Lurie. Another application is that the cohomology groups associated to any classical deformation problem (in any characteristic) admit the same operations as André-Quillen cohomology.

Cosimplicial DGLAs in Deformation Theory

“…Moreover, they prove that the functor MC sends surjective morphisms of DGLAs to fibrations of Kan complexes and quasi-isomorphisms of DGLAs to equivalences of Kan complexes; both these statements generalize to L -algebras. Since surjective morphisms and quasi-isomorphisms are the fibrations and the weak-equivalences in the standard model structure on the category of DGLAs (see [26]), Hinich and Getzler results are a first major step toward the formalization of the following folk statement: the 1 -category of L -algebras is equivalent to the 1 -category of formal -groupoids, which naturally generalizes the well-known equivalence between the category of Lie algebras and the category of formal Lie groups (in characteristic zero). A further major step in this direction has been made by Pridham, who proves Downloaded by [University of Chicago Library] at 20:12 16 November 2014 in [26] that the homotopy categories of DGLAs and of L -algebras are equivalent to a certain category of SSet-valued functors of Artin rings ('geometric' deformation functors, see also [24]).…”

The homotopy theory of strong homotopy algebras and bialgebras

“…Generating trivial cofibrations have the additional property that H * (coker f ) = 0. Characterisation of the weak equivalences follows from [Pri7,Proposition 4.42]. That β * is a Quillen equivalence follows from [Pri7,Theorem 4.55].…”

“…Existence of such a model structure is given in [Pri7,Proposition 4.36] for the analogous case of cocommutative coassociative coalgebras and Lie algebras, but the proof carries over to any Koszul-dual pair of quadratic operads, so it adapts to our context (coassociative coalgebras and associative algebras). The generating cofibrations are injective morphisms f : C → D between finite-dimensional objects, satisfying the additional property that the coproduct coker f → D ⊗ coker f is zero.…”

Derived deformations of Artin stacks

“…By [Pri4] Corollary 1.46, these have the property that for any simplicial k-vector space V with finite-dimensional normalisation,…”

“…With the exception of §1.4, the definitions and results in this section can all be found in [Pri4]. Fix a complete local Noetherian ring Λ, with maximal ideal µ and residue field k.…”