Subcategories of the category of L-convex spaces

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In this paper, several types of L-convex spaces are introduced, including stratified L-convex spaces, convex-generated L-convex spaces, weakly induced L-convex spaces and induced L-convex spaces. Their relations are discussed category-theoretically. Firstly, it is shown that there is a Galois correspondence between the category SL-CS of stratified L-convex spaces (resp. the category WIL-CS of weakly induced L-convex spaces) and the category L-CS of L-convex spaces. In particular, SL-CS and WIL-CS are both coreflective subcategories of L-CS. Secondly, it is proved that there is a Galois correspondence between the category CS of convex spaces and the category SL-CS (resp. WIL-CS). Specially, CS can be embedded into SL-CS and WIL-CS as a coreflective subcategory. Finally, it is shown that the category CGL-CS of convex-generated L-convex spaces, the category IL-CS of induced L-convex spaces and CS are isomorphic.