  In this paper we study the Lagrangian Floer theory over $\Z$ or $\Z_2$. Under
an appropriate assumption on ambient symplectic manifold, we show that the
whole story of Lagrangian Floer theory in \cite{fooo-book} can be developed
over $\Z_2$ coefficients, and over $\Z$ coefficients when Lagrangian
submanifolds are relatively spin. The main technical tools used for the
construction are the notion of the sheaf of groups, and stratification and
compatibility of the normal cones applied to the Kuranishi structure of the
moduli space of pseudo-holomorphic discs.
