  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.
