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.