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.
The purpose of this paper is to generalise Sullivan's rational homotopy theory to non-nilpotent spaces, providing an alternative approach to defining Toën's schematic homotopy types over any field k of characteristic zero. New features include an explicit description of homotopy groups using the Maurer-Cartan equations, convergent spectral sequences comparing schematic homotopy groups with cohomology of the universal semisimple local system, and a generalisation of the Baues-Lemaire conjecture. For compact Kähler manifolds, the schematic homotopy groups can be described explicitly in terms of this cohomology ring, giving them canonical weight decompositions. There are also notions of minimal models, unpointed homotopy types and algebraic automorphism groups. For a space with algebraically good fundamental group and higher homotopy groups of finite rank, the schematic homotopy groups are shown to be π n (X) ⊗ Z k.
We show that an n-geometric stack may be regarded as a special kind of
simplicial scheme, namely a Duskin n-hypergroupoid in affine schemes, where
surjectivity is defined in terms of covering maps, yielding Artin n-stacks,
Deligne-Mumford n-stacks and n-schemes as the notion of covering varies. This
formulation adapts to all HAG contexts, so in particular works for derived
n-stacks (replacing rings with simplicial rings). We exploit this to describe
quasi-coherent sheaves and complexes on these stacks, and to draw comparisons
with Kontsevich's dg-schemes. As an application, we show how the cotangent
complex controls infinitesimal deformations of higher and derived stacks.Comment: 55 pages; v3 content rearranged with many corrections; final version,
to appear in Adv. Math; v4 corrections in section 7.
Abstract. We show that on a derived Artin N -stack, there is a canonical equivalence between the spaces of n-shifted symplectic structures and non-degenerate n-shifted Poisson structures.
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