2020/06/02 by Behrend, Kai, Liao, Hsuan-Yi, Xu, Ping
#Algebraic Geometry (math.AG) #Algebraic Topology (math.AT) #Category Theory (math.CT) #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Theory (hep-th)
paper · doi:10.48550/arxiv.2006.01376
We develop the theory of derived differential geometry in terms of bundles of curved L_∞[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_∞[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.