Differential Graded Manifolds of Finite Positive Amplitude

Abstract

Abstract We prove that dg manifolds of finite positive amplitude, that is, 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.