In this paper, the notions of (L, M)-fuzzy concave spaces, (L, M)-fuzzy interior spaces, (L, M)-fuzzy interior relations and (L, M)-fuzzy hull relations are introduced. It is proved that the category of (L, M)-fuzzy concave spaces, the category of (L, M)-fuzzy interior spaces, the category of (L, M)-fuzzy interior relation spaces and the category of (L, M)-fuzzy hull relation spaces are isomorphic. Moreover, it is proved that these categories are all isomorphic to the category of (L, M)-fuzzy convex spaces when L is a completely distributive lattice with an order-reversing involution.
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