# Derived smooth manifolds

@article{Spivak2008DerivedSM, title={Derived smooth manifolds}, author={David I. Spivak}, journal={Duke Mathematical Journal}, year={2008}, volume={153}, pages={55-128} }

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…

## 57 Citations

Derived smooth stacks and prequantum categories

- Mathematics
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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

- MathematicsCommunications in Mathematical Physics
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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

- Mathematics
- 2019

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

- Mathematics
- 2012

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

- Mathematics
- 2012

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

- Mathematics
- 2012

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

- Mathematics
- 2020

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

- MathematicsVirtual 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

- Mathematics
- 2015

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.

- Mathematics
- 2011

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…

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