Categories and General Algebraic Structures with Applications

Categories and General Algebraic Structures with Applications

Doctrinal soft structures and Molodtsov equivalence: A categorical reconstruction of correct soft operations

Articles in Press

Document Type : Research Paper

Author
Mustafa Burç Kandemir
Department of Mathematics, Mu˘gla Sıtkı Ko¸cman University, Mu˘gla, T¨urkiye
10.48308/cgasa.2026.245396.1651
Abstract
Molodtsov's soft set theory represents uncertainty by assigning subsets to parameters, but many operations on soft sets may depend on the chosen parameter names rather than on the represented family of subsets. Motivated by Molodtsov's equivalence and correct-operation principles, this paper gives a categorical reconstruction of soft structures over an indexed poset $P:\mathcal C^{op}\to \mathbf{Pos}$. A parameterized soft object \(S:A\to P(X)\) is treated as a representative, while its invariant content is the image family \(\operatorname{Im}S\subseteq P(X)\). We define doctrinal soft structures as such fibrewise families and show that they form the Grothendieck construction of a naturally induced indexed preorder. The canonical projection to \(\mathcal C\) is proved to be a Grothendieck fibration. We also characterize correct soft operations as reindexing-compatible fibrewise operators, construct pullbacks over base pullbacks under finite-meet assumptions, and extend Molodtsov's internal and external approximation equivalences to finite poset fibres via minimal and maximal antichain representatives.
Keywords
Soft set
Molodtsov equivalence
correct operation
indexed poset
doctrine
Grothendieck construction
fibration
subobject doctrine
Subjects
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Categories and General Algebraic Structures with Applications

Articles in Press, Accepted Manuscript
Available Online from 28 September 2026