  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.
