TY - JOUR T1 - The categories of $L$-convex spaces and $L$-convergence spaces: extensionality and productivity of quotient maps AU - Pang, Bin AU - Han, Xiancheng PY - 2025 DA - August Y2 - 2024 DO - 10.15672/hujms.1543634 JF - Hacettepe Journal of Mathematics and Statistics PB - Hacettepe University WT - DergiPark SN - 2651-477X SP - 1257 EP - 1275 VL - 54 IS - 4 LA - en AB - 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. 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