Coreflective subcategories and classical embedding in L-measurable spaces category

Year 2026, Issue: Advanced Online Publication, 30.12.2025

Xiu Xin , Huan Lian

https://doi.org/10.15672/hujms.1784919

Abstract

To meet the measurement needs in scenarios with fuzziness and
uncertainty, fuzzy measure theory needs to extend the framework of
classical measure theory to the lattice-valued context. However,
existing studies have limitations such as difficulties in verifying
the closedness of complement operations and insufficient connections
at the categorical level. Focusing on Shi$'$s
\(L\)-\(\sigma\)-algebras and combining cut set theory with category
theory, this paper defines subclasses of \(L\)-measurable spaces and
constructs the relevant categorical framework. By using Galois
correspondence and closure functors, it proves that the category of
stratified \(L\)-measurable spaces \(L\)-{\bf SMS} and the category
of weakly induced \(L\)-measurable spaces \(L\)-{\bf WIMS} are both
coreflective full subcategories of the category of \(L\)-measurable
spaces \(L\)-{\bf MS}. Based on cut sets, it establishes
embedding/reduction functors, confirming that the category of
classical measurable spaces {\bf MS} can be embedded into \(L\)-{\bf
SMS} and \(L\)-{\bf WIMS}, and is isomorphic to the category of
generated \(L\)- \(\sigma\)-algebras and the category of induced
\(L\)-measurable spaces \(L\)-{\bf IMS}. This research addresses the
defects of traditional frameworks, improves the categorical system
of lattice-valued measurable spaces, and provides support for the
cross-application of the two types of measure theories.

Keywords

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

References

• [1] J. Ad´amek, H. Herrlich, G.E. Strecker, Abstract and Concrete Categories, Wiley, New York, 1990.