Let $A=\bigoplus_{ p\in G}A_{ p}$ be a multiplier Hopf $T$-coalgebra over a group $G$, in this paper we give the definition of the crossed left $A$-$G$-modules and show that the category of crossed left $A$-$G$-modules is a monoidal category. Finally we show that a family of multipliers $R = \{ R_{ p, q} \in M(A_{ p}\otimes A_{ q})\}_{ p, q\in G}$ is a quasitriangular structure of a multiplier $T$-coalgebra $A$ if and only if the crossed left $A$-$G$-module category over $A$ is a braided monoidal category with the braiding $c$ defined by $R$, generalizing the main results in \cite{ ZCL11} to the more general framework of multiplier Hopf algebras.
Categories: math.RA