Derived smooth manifolds

  title={Derived smooth manifolds},
  author={David I. Spivak},
  journal={Duke Mathematical Journal},
  • David I. Spivak
  • Published 29 October 2008
  • Mathematics
  • Duke Mathematical Journal
We define a simplicial category called the category of derived manifolds. It contains the category of smooth manifolds as a full discrete subcategory, and it is closed under taking arbitrary intersections in a manifold. A derived manifold is a space together with a sheaf of local $C^\infty$-rings that is obtained by patching together homotopy zero-sets of smooth functions on Euclidean spaces. We show that derived manifolds come equipped with a stable normal bundle and can be imbedded into… 
Derived smooth stacks and prequantum categories
The Weil-Kostant integrality theorem states that given a smooth manifold endowed with an integral complex closed 2-form, then there exists a line bundle with connection on this manifold with
Dg Manifolds, Formal Exponential Maps and Homotopy Lie Algebras
This paper is devoted to the study of the relation between ‘formal exponential maps,’ the Atiyah class, and Kapranov L∞[1] algebras associated with dg manifolds in the C ∞ context. We prove that, for
Structured Brown representability via concordance
We establish a highly structured variant of the Brown representability theorem: given a sheaf of spaces on the site of manifolds, we show that concordance classes of sections of this sheaf over a
On manifolds with corners
Manifolds without boundary, and manifolds with boundary, are universally known in Differential Geometry, but manifolds with corners (locally modelled on [0,\infty)^k x R^{n-k}) have received
Moduli Spaces: An introduction to d-manifolds and derived differential geometry
This is a survey of the author's book "D-manifolds and d-orbifolds: a theory of derived differential geometry", available at this http URL We introduce a 2-category dMan of "d-manifolds", new
D-manifolds, d-orbifolds and derived differential geometry: a detailed summary
This is a long summary of the author's book "D-manifolds and d-orbifolds: a theory of derived differential geometry", available at this http URL . A shorter survey paper on the book, focussing on
Derived Differential Geometry
We develop the theory of derived differential geometry in terms of bundles of curved $L_\infty[1]$-algebras, i.e. dg manifolds of positive amplitudes. We prove the category of derived manifolds is a
Kuranishi spaces as a 2-category
  • D. joyce
  • Mathematics
    Virtual Fundamental Cycles in Symplectic Topology
  • 2019
This is a survey of the author's in-progress book arXiv:1409.6908. 'Kuranishi spaces' were introduced in the work of Fukaya, Oh, Ohta and Ono in symplectic geometry (see e.g. arXiv:1503.07631), as
Derived complex analytic geometry I: GAGA theorems
We further develop the foundations of derived complex analytic geometry introduced in [DAG-IX] by J. Lurie. We introduce the notion of coherent sheaf on a derived complex analytic space. Moreover,
Stacks and their function algebras.
For T any abelian Lawvere theory, we establish a Quillen adjunction between model category structures on cosimplicial T -algebras and on simplicial presheaves over duals of T -algebras, whose left


Homotopical Algebraic Geometry II: Geometric Stacks and Applications
This is the second part of a series of papers devoted to develop Homotopical Algebraic Geometry. We start by defining and studying generalizations of standard notions of linear and commutative
Convenient Categories of Smooth Spaces
A "Chen space" is a set X equipped with a collection of "plots" - maps from convex sets to X - satisfying three simple axioms. While an individual Chen space can be much worse than a smooth manifold,
Algebraic theories in homotopy theory
It is well known in homotopy theory that given a loop space X one can always find a simplicial group G weakly equivalent to X, such that the weak equivalence can be realized by maps preserving
Derived Hilbert schemes
We construct the derived version of the Hilbert scheme parametrizing subschemes in a given projective scheme X with given Hilbert polynomial h. This is a dg-manifold (smooth dg-scheme) RHilb_h(X)
Models for smooth infinitesimal analysis
The aim of this book is to construct categories of spaces which contain all the C?-manifolds, but in addition infinitesimal spaces and arbitrary function spaces. To this end, the techniques of
Handbook of Categorical Algebra
The Handbook of Categorical Algebra is intended to give, in three volumes, a rather detailed account of what, ideally, everybody working in category theory should know, whatever the specific topic of
Every homotopy theory of simplicial algebras admits a proper model