Using a base-point free version of the infinite symmetric product we define a fibrewise infinite symmetric product for any fibration $E;longrightarrow;B$. The construction works for any commutative ring R with unit and is denoted by $R_f(E);longrightarrow;B$. For any pointed space B let $G_I(B);longrightarrow;B$ be the i-th Ganea fibration. Defining $M_R-cat(B):= inf{imidR_f(G_i(B))longrihghtarrow;B$ admits a section} we obtain an approximation to the Lusternik-Schnirelmann category of B which satisfies .g.a product formula. In particular, if B is a 1-connected rational space of finite rational type, then $M_Q$-cat(B) coincides with the well-known (purely algebraically defined) M-category of B which in fact is equal to cat (B) by a result of K.Hess. All the constructions more generally apply to the Ganea category of maps.