Soft set theory has been developed as an effective mathematical tool for handling uncertainty through parameterization, leading to more practical solutions to complex problems. In parallel, the theory of covering spaces and their algebraic interpretation via groupoids has played a fundamental role in algebraic topology. Previous studies have established important categorical equivalences between coverings of spaces, coverings of groupoids, and groupoid actions on a set. In this paper, these ideas are extended to the framework of soft set theory. We introduce the notion of soft covering groupoids and define the category $ SGdCov(H) $ of soft coverings of a soft groupoid $ H $. Additionally, we define the actions of soft groupoids and the category $ SGdOp(H) $ of soft actions of soft groupoid $ H $. The main result of this study is the proof of a categorical equivalence between these two categories, thereby providing a new algebraic perspective on soft covering structures and contributing to the interaction between soft set theory and categorical topology.
Citation: Semih Geçen, İlhan İçen, Abdullah Fatih Özcan. Covering morphisms of soft groupoids[J]. AIMS Mathematics, 2026, 11(4): 11372-11386. doi: 10.3934/math.2026468
Soft set theory has been developed as an effective mathematical tool for handling uncertainty through parameterization, leading to more practical solutions to complex problems. In parallel, the theory of covering spaces and their algebraic interpretation via groupoids has played a fundamental role in algebraic topology. Previous studies have established important categorical equivalences between coverings of spaces, coverings of groupoids, and groupoid actions on a set. In this paper, these ideas are extended to the framework of soft set theory. We introduce the notion of soft covering groupoids and define the category $ SGdCov(H) $ of soft coverings of a soft groupoid $ H $. Additionally, we define the actions of soft groupoids and the category $ SGdOp(H) $ of soft actions of soft groupoid $ H $. The main result of this study is the proof of a categorical equivalence between these two categories, thereby providing a new algebraic perspective on soft covering structures and contributing to the interaction between soft set theory and categorical topology.
| [1] |
D. Molodtsov, Soft set theory–First results, Comput. Math. Appl., 37 (1999), 19–31. https://doi.org/10.1016/S0898-1221(99)00056-5 doi: 10.1016/S0898-1221(99)00056-5
|
| [2] |
H. Aktaş, N. Çaǧman, Soft sets and soft groups, Inform. Sci., 177 (2007), 2726–2735. https://doi.org/10.1016/j.ins.2006.12.008 doi: 10.1016/j.ins.2006.12.008
|
| [3] |
U. Acar, F. Koyuncu, B. Tanay, Soft sets and soft rings, Comput. Math. Appl., 59 (2010), 3458–3463. https://doi.org/10.1016/j.camwa.2010.03.034 doi: 10.1016/j.camwa.2010.03.034
|
| [4] | G. Oguz, İ. İçen, M. H. Gursoy, A new concept in the soft theory: soft groupoids, Southeast Asian Bull. Math., 44 (2020), 555–565. |
| [5] |
M. Shabir, M. Naz, On soft topological spaces, Comput. Math. Appl., 61 (2011), 1786–1799. https://doi.org/10.1016/j.camwa.2011.02.006 doi: 10.1016/j.camwa.2011.02.006
|
| [6] | S. Gecen, Reflections of global action on soft set theory, MS. Thesis, İnönü University, 2021. |
| [7] | G. Oguz, M. H. Gursoy, İ. İçen, On soft topological categories, Hacettepe J. Math. Stat., 48 (2019), 1675–1681. |
| [8] | S. K. Sardar, S. Gupta, Soft category theory-an introduction, J. Hyperstructures, 2 (2013), 118–135. |
| [9] |
R. Brown, O. Mucuk, Covering groups of non-connected topological groups revisited, Math. Proc. Cambridge Philos. Soc., 115 (1994), 97–110. https://doi.org/10.1017/S0305004100071942 doi: 10.1017/S0305004100071942
|
| [10] | M. H. Gursoy, İ. İçen, Coverings of structured Lie groupoids, Hacettepe J. Math. Stat., 47 (2018), 845–854. |
| [11] | O. Mucuk, İ. İçen, Coverings of groupoids, Hadronic J. Suppl., 16 (2001), 183–196. |
| [12] | R. Brown, Topology and groupoids, A Geometric Account of General Topology, Homotopy Types and the Fundamental Groupoid, Deganwy: BookSurge LLC, 2006. |
| [13] | P. Gabriel, M. Zisman, Calculus of fractions and homotopy theory, Vol. 35, Springer-Verlag, 1967. https://doi.org/10.1007/978-3-642-85844-4 |
| [14] |
R. Brown, G. Danesh-Naruie, J. P. L. Hardy, Topological groupoids: Ⅱ. Covering morphisms and $G$-spaces, Math. Nachr., 74 (1976), 143–156. https://doi.org/10.1002/mana.3210740110 doi: 10.1002/mana.3210740110
|
| [15] |
A. F. Ozcan, İ. İçen, Topological group-groupoids and equivalent categories, Yüzüncü Yıl Üniv. Fen Bilimleri Enstitüsü Dergisi, 27 (2022), 667–673. https://doi.org/10.53433/yyufbed.1137668 doi: 10.53433/yyufbed.1137668
|
| [16] |
İ. İçen, M. H. Gursoy, A. F. Ozcan, Coverings of Lie groupoids, Turkish J. Math., 35 (2011), 207–218. https://doi.org/10.3906/mat-0902-2 doi: 10.3906/mat-0902-2
|
| [17] |
A. F. Ozcan, İ. İçen, Actions of $R$-module groupoids, Filomat, 37 (2023), 4239–4247. https://doi.org/10.2298/FIL2313239O doi: 10.2298/FIL2313239O
|
| [18] |
P. K. Maji, R. Biswas, A. R. Roy, Soft set theory, Comput. Math. Appl., 45 (2003), 555–562. https://doi.org/10.1016/S0898-1221(03)00016-6 doi: 10.1016/S0898-1221(03)00016-6
|
| [19] |
G. Oguz, İ. İçen, M. H. Gursoy, Actions of soft groups, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., 68 (2019), 1163–1174. https://doi.org/10.31801/cfsuasmas.509926 doi: 10.31801/cfsuasmas.509926
|
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Semih Geçen, İlhan İçen, Abdullah Fatih Özcan. Covering morphisms of soft groupoids[J]. AIMS Mathematics, 2026, 11(4): 11372-11386. doi: 10.3934/math.2026468
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