The categories of $L$-convex spaces and $L$-convergence spaces: extensionality and productivity of quotient maps

Based on a complete residuated lattice $L$, we show that the category of $L$-convex spaces is not extensional and is closed under the formation of finite products of quotient maps. Then we propose the concept of (preconcave, concave) $L$-convergence spaces via $L$-co-Scott closed sets and prove that the category of concave $L$-convergence spaces is isomorphic to that of $L$-concave spaces. Finally, we investigate the categorical properties of $L$-convergence spaces and show that it is extensional and closed under the formation of finite products of quotient maps.