The category of X-decomposable ordered vector spaces over a convenient category X is introduced and it is shown to uphold an exponential law [E8F,G] = [E,[F, G]] and to have limits and colimits. As the main special cases of X, the categories Cv (convergence spaces) and Bo (bornological spaces) are considered. The topological Cv-decomposable ordered vector spaces are precisely the ordered topological vector spaces whose positive cones give open decompositions.