Exponentiable Objects and Function spaces in Lowen Fuzzy Topological Spaces

We study exponentiable objects and function spaces in the category of stratified Lowen fuzzy topological spaces over $\text{I}=[0,1]$. Using the Lowen fuzzy Sierpiński object $\text{Sier}$, which identifies ${\tau }_X$ with $C(X,\text{Sier})$, we explicitly determine the largest splitting topology on this mapping set. Its open weights $\mathrm{\Phi }:{\tau }_X\to \text{I}$ are precisely those satisfying Scott continuity and a finite-tier compatibility condition induced by finite powers of $\text{Sier}$. This yields an intrinsic characterization: $X$ is exponentiable if and only if every $\mu \in {\tau }_X$ satisfies $$\mu =\underset{\lambda ◃\mathrm{\Phi }}{\bigvee }(\text{const}\mathrm{\Phi }(\mu )\wedge \lambda ),\lambda ◃\mathrm{\Phi }⟺\text{const}\mathrm{\Phi }(\nu )\wedge \lambda \le \nu (\nu \in {\tau }_X)\mathrm{.}$$ When this condition holds, $Y^X$ has underlying set $C(X,Y)$, with topology generated by $[\mathrm{\Phi },v](f)=\mathrm{\Phi }(v\circ f)$. We also obtain a dual closed-set formulation and three applications. Exponentiability implies that ${\tau }_X$ is a continuous lattice, although the converse fails. Moreover, a classical space $X$ is exponentiable exactly when its induced fuzzy space $\omega X$ is exponentiable in the entire stratified Lowen category. Finally, Lowen compact, strongly fuzzy compact, and $N$-compact Hausdorff spaces are exponentiable.