Homotopy L-infinity spaces and Kuranishi manifolds, I: categorical structures

Motivated by the definition of homotopy $L_\infty$ spaces, we develop a new theory of Kuranishi manifolds, closely related to Joyce's recent theory. We prove that Kuranishi manifolds form a $2$-category with invertible $2$-morphisms, and that certain fiber product property holds in this $2$-category. In a subsequent paper, we construct the virtual fundamental cycle of a compact oriented Kuranishi manifold, and prove some of its basic properties. Manifest from this new formulation is the fact that $[0,1]$-type homotopy $L_\infty$ spaces are naturally Kuranishi manifolds. The former structured spaces naturally appear as derived enhancements of Maurer-Cartan moduli spaces from Chern-Simons type gauge theory. In this way, Kuranishi manifolds theory can be applied to study path integrals in such type of gauge theories.