From Hag To Dag: Derived Moduli Stacks

By Bertrand Toën, G. Vezzosi. 2002

80 citations recorded by the source corpus.

These are expanded notes of some talks given during the fall 2002, about homotopical algebraic geometry with special emphasis on its applications to derived algebraic geometry and derived deformation theory. We use the general framework developed in [HAG-I], and in particular the notions of model topology, model sites and stacks over them, in order to define various derived moduli functors and study their geometric properties. We start by defining the model category of D-stacks, with respect to an extension of the etale topology to the category of commutative differential graded algebras, and we show that its homotopy category contains interesting objects, such as schemes, algebraic stacks, higher algebraic stacks, dg-schemes, etc. We define the notion of geometric D-stacks and present some related geometric constructions (O-modules, perfect complexes, K-theory, derived tangent stacks, cotangent complexes, various notions of smoothness, etc.). Finally, we define and study the derived moduli problems classifying local systems on a topological space, vector bundles on a smooth projective variety, and A ∞-categorical structures. We state geometricity and smoothness results for these examples. The proofs of the results presented in this paper will be mainly given in [HAG-II].

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/fb050bcea7aea91e143de837c346683a79304fcf

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In Mathematical Physics

Citations in the corpus

Homotopical algebraic geometry. I. Topos theory. (2002, methodology): “We refer the reader to the Introduction of the research announcement [To-Ve 5] and to [To-Ve 4], where motivations and prospective applications (mainly to the so-calledderived moduli spacesof [Ko,Ci-Ka1,Ci-Ka2]) are provided.”
Derived Algebraic Geometry (2014, background): “They date mostly from the first decade of this century and appear in a series of works: [136], [137], [138], [90], [128], [91].”
Derived Algebraic Geometry V: Structured Spaces (2009, background): “For another very general foundational approach (of a rather different flavor from ours), we refer the reader to [19] and [20].”
Non-Archimedean analytic geometry as relative algebraic geometry (2013, methodology): “We are developing a formalism of derived analytic stacks along the lines of following Toën and Vezzosi’s approach [58] or Lurie’s approach [29, 30].”
Higher and derived stacks: a global overview (2009, background): “As explained in [To-Ve3] it is possible to construct a functor”