Lattice representation of L-quasi-convex spaces

Based on a complete Heyting algebra L, we first propose the concept of L-quasi-convex spaces and construct an adjunction between the category of L-S₀-quasi-convex spaces and the opposite category of complete L-ordered sets. Then we present the concept of weakly fuzzy algebraic lattices and prove that an L-quasi-convex structure endowed with the fuzzy inclusion order is precisely a weakly fuzzy algebraic lattice. Secondly, we introduce the notion of sobriety in L-quasi-convex spaces from the perspective of categorical equivalence, showing that the category of sober L-quasi-convex spaces is dually equivalent to that of weakly fuzzy algebraic lattices. Finally, we construct a monad on the category of L-S₀-quasi-convex spaces and obtain that the EilenbergMoore algebras of this monad are precisely sober L-quasi-convex spaces.