Crossed modules are algebraic structures that generalize the concept of group extensions. They involve group-like objects (often groups or groupoids) with additional structure and mappings between them that satisfy certain properties. Generalized crossed modules further extend this concept to higher-dimensional settings or more general algebraic contexts. In this paper, we studied the fibration and co-fibration of generalized crossed modules.
Citation: Hatice Gülsün Akay. (Co-)fibration of generalized crossed modules[J]. AIMS Mathematics, 2024, 9(11): 32782-32796. doi: 10.3934/math.20241568
Crossed modules are algebraic structures that generalize the concept of group extensions. They involve group-like objects (often groups or groupoids) with additional structure and mappings between them that satisfy certain properties. Generalized crossed modules further extend this concept to higher-dimensional settings or more general algebraic contexts. In this paper, we studied the fibration and co-fibration of generalized crossed modules.
| [1] |
A. Aytekin, K. Emir, Colimits of crossed modules in modified categories of interest, Electron. Res. Arch., 28 (2020), 1227–1238. http://dx.doi.org/10.3934/era.2020067 doi: 10.3934/era.2020067
|
| [2] |
A. Aytekin, (Co) Limits of Hom-Lie crossed module, Turk. J. Math., 45 (2021), 2140–2153. https://doi.org/10.3906/mat-2106-43 doi: 10.3906/mat-2106-43
|
| [3] | R. Brown, Topology and groupoids, Carolina: Booksurge LLC, 2006. |
| [4] |
R. Brown, P. J. Higgins, On the connection between the second relative homotopy groups of some related spaces, P. Lond. Math. Soc., 3 (1978), 193–212. https://doi.org/10.1112/plms/s3-36.2.193 doi: 10.1112/plms/s3-36.2.193
|
| [5] |
R. Brown, R. Sivera, Algebraic colimit calculations in homotopy theory using fibred and cofibred categories, arXiv, 22 (2009), 222–251. https://doi.org/10.48550/arXiv.0809.4192 doi: 10.48550/arXiv.0809.4192
|
| [6] |
J. M. Casas, M. Ladra, Colimits in the crossed modules category in Lie algebras, Georgian Math. J., 7 (2000), 461–474. https://doi.org/10.1515/GMJ.2000.461 doi: 10.1515/GMJ.2000.461
|
| [7] |
U. Ege Arslan, İ. İ. Akça, G. Onarlı Irmak, O. Avcıoğlu, Fibrations of 2-crossed modules, Math. Method. Appl. Sci., 42 (2019), 5293–5304. https://doi.org/10.1002/mma.5321 doi: 10.1002/mma.5321
|
| [8] |
K. Emir, S. Çetin, Limits in modified categories of interest, arXiv, 43 (2017), 2617–2634. https://doi.org/10.48550/arXiv.1805.04877 doi: 10.48550/arXiv.1805.04877
|
| [9] | J. W. Gray, Fibred and cofibred categories, In: Proceedings of the conference on categorical algebra, Berlin: Springer Berlin Heidelberg, 1996. |
| [10] | A. Grothendieck, Catégories fibrées et descente, Seminaire de géométrie algébrique de l'Institut des Hautes Études Scientifiques, Paris, 1961. |
| [11] |
Ö. Gürmen Alansal, U. Ege Arslan, Crossed modules bifibred over k-Algebras, Cumhuriyet Science Journal, 42 (2021), 99–114. https://doi.org/10.17776/csj.727906 doi: 10.17776/csj.727906
|
| [12] |
E. Soylu Yilmaz, (Co)Limit calculations in the category of 2-crossed $R$-modules, Turk. J. Math., 46 (2022), 2902–2915. https://doi.org/10.55730/1300-0098.3308 doi: 10.55730/1300-0098.3308
|
| [13] | J. H. C. Whitehead, Combinatorial homotopy Ⅰ, Homotopy Theory, 1962, 85–117. https://doi.org/10.1016/B978-0-08-009871-5.50012-X |
| [14] | J. H. C. Whitehead, Combinatorial homotopy Ⅱ, Homotopy Theory, 1962,119–162. https://doi.org/10.1016/B978-0-08-009871-5.50013-1 |
| [15] |
M. Yavari, A. Salemkar, The category of generalized crossed modules, Categ. Gen. Algebr. Struct. Appl., 10 (2019), 157–171. https://doi.org/10.29252/CGASA.10.1.157 doi: 10.29252/CGASA.10.1.157
|
Hatice Gülsün Akay. (Co-)fibration of generalized crossed modules[J]. AIMS Mathematics, 2024, 9(11): 32782-32796. doi: 10.3934/math.20241568
DownLoad: