Two types of Galois correspondences over quantaloid-typed sets

Abstract

For a small, integral and meet-continuous quantaloid Q, we establish two types of Galois correspondences by considering a limit structure on a set as a Q-multiple limit structure on a Q-typed set. One Galois  correspondence shows that the stratied Q-topologies and the Q-multiple limit structures based on Q-typed sets can be converted to each other categorically. Moreover, the other one shows that a new mathematical structure on a Q-typed set, namely ⊤-Q-limit structure, can be obtained from Q-multiple limit structures categorically. In the case that Q is the quantaloid D(L) of diagonals obtained from a GL-quantale L, the rst Galois  correspondence captures a pair of concrete functors between the concrete category of D(L)-multiple limit spaces and that of stratied D(L)-topological spaces over the slice category Set#L, and the second Galois correspondence captures a pair of concrete functors between the concrete category of D(L)-multiple limit spaces and that of ⊤-D(L)-limit spaces over the slice category Set#L. Last but not the least, the rst Galois correspondence reduces to the famous Lowen's functors, while the second Galois correspondence captures a new relationship between the construct of limit spaces and that of ⊤-limit spaces in the case that the  underlying quantale L is considered as a one-object quantaloid.