Stratified $L$-convex groups

This paper investigates a novel structure of stratified $L$-convex groups, defined as groups possessing stratified $L$-convex structures, in which the group operations are $L$-convexity-preserving mappings. It is verified that stratified $L$-convex groups serve as objects, while $L$-convexity-preserving group homomorphisms serve as morphisms, together forming a concrete category, denoted as SLCG. As a specific instance of SLCG (i.e., when $L$=2), the category of convex groups, denoted as CG, is also defined. We show that CG can be embedded within SLCG as a reflective subcategory. In addition, we demonstrate that SLCG possesses well-defined characterizations, localization properties, and initial and final structures, establishing it as a topological category over groups.