Two types of Galois correspondences over quantaloid-typed sets. Issue 6 (3rd June 2021)

• Record Type: Journal Article
• Authors: Fang, Jinming; Fang, Zhou
• Is Part Of: Quaestiones mathematicae. Volume 44:Issue 6(2021)
• Page Start: 765
• Page End: 792
• Publication Date: 2021-06-03
• Subjects: 54A40 -- 54A20 -- 06F07 -- 94D05; Quantaloid-typed set -- limit space -- quantaloid-multiple limit space -- quantaloid-enriched topology -- top quantaloid-enriched limit structure; Mathematics -- Periodicals; 510.5
• DOI: 10.2989/16073606.2020.1746429
• Languages: English
• ISSNs: 1607-3606

Abstract

Abstract: For a small, integral and meet-continuous quantaloid, we establish two types of Galois correspondences by considering a limit structure on a set as a -multiple limit structure on a -typed set. One Galois correspondence shows that the stratified -topologies and the -multiple limit structures based on -typed sets can be converted to each other categorically. Moreover, the other one shows that a new mathematical structure on a -typed set, namely ⊤ - -limit structure, can be obtained from -multiple limit structures categorically. In the case that is the quantaloid ( L ) of diagonals obtained from a GL-quantale L, the first Galois correspondence captures a pair of concrete functors between the concrete category of ( L )-multiple limit spaces and that of stratified ( 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 ( L )-multiple limit spaces and that of ⊤ - ( L )-limit spaces over the slice category Set ↓ L . Last but not the least, the first 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.