Derived Differentiable Manifolds

TL;DR

This paper develops the theory of derived differential geometry using curved $L_
[1]$-algebras, establishing a homotopy-theoretic framework for derived manifolds and their intersections.

Contribution

It introduces a new categorical framework for derived manifolds, including homotopy fibered products and derived intersections, using path space factorizations and the homotopy transfer theorem.

Findings

Category of derived manifolds is a category of fibrant objects

Constructed homotopy fibered products and derived intersections

Proved inverse function theorem for derived 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.