Coreflective subcategories and classical embedding in \(L\)-measurable spaces category

Abstract

To meet the measurement needs in scenarios with fuzziness anduncertainty, fuzzy measure theory needs to extend the framework ofclassical measure theory to the lattice-valued context. However,existing studies have limitations such as difficulties in verifyingthe closedness of complement operations and insufficient connectionsat the categorical level. Focusing on Shi$'$s\(L\)-\(\sigma\)-algebras and combining cut set theory with categorytheory, this paper defines subclasses of \(L\)-measurable spaces andconstructs the relevant categorical framework. By using Galoiscorrespondence and closure functors, it proves that the category ofstratified \(L\)-measurable spaces \(L\)-{\bf SMS} and the categoryof weakly induced \(L\)-measurable spaces \(L\)-{\bf WIMS} are bothcoreflective full subcategories of the category of \(L\)-measurablespaces \(L\)-{\bf MS}. Based on cut sets, it establishesembedding/reduction functors, confirming that the category ofclassical measurable spaces {\bf MS} can be embedded into \(L\)-{\bfSMS} and \(L\)-{\bf WIMS}, and is isomorphic to the category ofgenerated \(L\)-\(\sigma\)-algebras and the category of induced\(L\)-measurable spaces \(L\)-{\bf IMS}. This research addresses thedefects of traditional frameworks, improves the categorical systemof lattice-valued measurable spaces, and provides support for thecross-application of the two types of measure theories.

Keywords

• \(L\)-$\sigma$-algebra
• Galois correspondence
• coreflective subcategory
• cut set theory
• classical embedding

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