Two types of Galois correspondences over quantaloid-typed sets

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.

Mathematics Subject Classification (2010):

• 54A40
• 54A20
• 06F07
• 94D05

Key words:

• Quantaloid-typed set
• limit space
• quantaloid-multiple limit space
• quantaloid-enriched topology
• top quantaloid-enriched limit structure