Differential graded manifolds of finite positive amplitude

TL;DR

This paper establishes a category of fibrant objects for dg manifolds of finite positive amplitude, using path space constructions and homotopy transfer theorems, with applications to derived intersections of manifolds.

Contribution

It introduces a new categorical framework for dg manifolds of finite positive amplitude and develops a factorization theorem using path spaces and homotopy transfer techniques.

Findings

Proves dg manifolds of finite positive amplitude form a category of fibrant objects.

Constructs an infinite-dimensional factorization of the diagonal morphism via path spaces.

Applies the theory to study derived intersections of manifolds.

Abstract

We prove that dg manifolds of finite positive amplitude, i.e. bundles of positively graded curved $L_\infty[1]$-algebras, form a category of fibrant objects. As a main step in the proof, we obtain a factorization theorem using path spaces. First we construct an infinite-dimensional factorization of a diagonal morphism using actual path spaces motivated by the AKSZ construction. Then we cut down to finite dimensions using the Fiorenza-Manetti method. The main ingredient in our method is the homotopy transfer theorem for curved $L_\infty[1]$-algebras. As an application, we study the derived intersections of manifolds.