  In this paper we will define an invariant $mc_{\infty}(f)$ of maps $f:X
\rightarrow Y_{\mathbb{Q}}$ between a finite CW-complex and a rational space
$Y_{\mathbb{Q}}$. We prove that this invariant is complete, i.e.
$mc_{\infty}(f)=mc_{\infty}(g)$ if an only if $f$ and $g$ are homotopic. We
will also construct an $L_{\infty}$-model for the based mapping space
$Map_*(X,Y_{\mathbb{Q}})$ from a $C_{\infty}$-coalgebra and an
$L_{\infty}$-algebra.
