Derived Differentiable Manifolds

Abstract

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 category of fibrant objects. Therefore, we can make sense of "homotopy fibered product" and "derived intersection" of submaifolds in a smooth manifold in the homotopy category of derived manifolds. We construct a factorization of the diagonal using path spaces. First we construct an infinite-dimensional factorization using actual path spaces motivated by the AKSZ construction, then we cut down to finite dimensions using the Fiorenza-Manetti method. The main ingredient is the homotopy transfer theorem for curved $L_{\infty }[1]$-algebras. We also prove the inverse function theorem for derived manifolds, and investigate the relationship between weak equivalence and quasi-isomorphism for derived manifolds.

Minor revision; references added; added a subsection about homotopy fibered products and derived intersections

References in corpus (8)

• On the perturbation lemma, and deformations
• Supergroupoids, double structures, and equivariant cohomology
• The Atiyah class of a dg-vector bundle
• Differential Graded Schemes I: Perfect Resolving Algebras
• Differential Graded Schemes II: The 2-category of Differential Graded Schemes
• Homological Algebra for Superalgebras of Differentiable Functions
• Differential gorms, differential worms
• The inverse function theorem for curved L-infinity spaces

Cited by in corpus (2)

• Dg manifolds, formal exponential maps and homotopy Lie algebras
• The inverse function theorem for curved L-infinity spaces