Extensional subobjects in categories of $\Omega$-fuzzy sets

summary:Two categories $\mathbb{Set}(\Omega )$ and $\mathbb{SetF}(\Omega )$ of fuzzy sets over an $MV$-algebra $\Omega $ are investigated. Full subcategories of these categories are introduced consisting of objects $(\mathop {{\mathrm sub}}(A,\delta )$, $\sigma )$, where $\mathop {{\mathrm sub}}(A,\delta )$ is a subset of all extensional subobjects of an object $(A,\delta )$. It is proved that all these subcategories are quasi-reflective subcategories in the corresponding categories