Research Article
BibTex RIS Cite

Completeness of the Category of Rack Crossed Modules

Year 2022, Volume: 4 Issue: 2, 56 - 68, 31.12.2022
https://doi.org/10.54286/ikjm.1199290

Abstract

In this paper, we prove that the category of rack crossed modules (with a fixed codomain) is finitely complete. In other words, we construct the product, pullback and equalizer objects in the category of crossed modules of racks. We therefore unify the group-theoretical analogy of the completeness property in the sense of the functor $\mathbf{Conj \colon Grp \to Rack} $.

References

  • Alp, M. (1998) Pullbacks of crossed modules and cat-1 groups. Turkish Journal of Mathematics, 22:273-281.
  • Alp, M. (2006) Pullbacks of crossed modules and cat-1 commutative algebras, Turkish Journal of Mathematics, 30:237-246.
  • Brown, R., Wensley, C.D. (1995) On infinite induced crossed modules and the homotopy 2-type of mapping cones. Theory Appl. Categ., 1.
  • Crans, A.S., Wagemann, F. (2014) Crossed modules of racks. Homology Homotopy Appl., 16(2):85–106.
  • Davvaz, B., Emir, K. (2020) On morphisms of crossed polymodules. Notices of AMS, 9(1):95-103.
  • Ege Arslan, U., Onarlı, G., (2015) An embedding theorem for the category of crossed p-modules. Georgian Mathematical Journal, 22(3):361–371.
  • Emir, K., Gülsün Akay, H. (2019) Pullback crossed modules in the category of racks. Hacettepe Journal of Mathematics and Statistics, 48(1):140-149.
  • Emir, K., Çetin, S. (2017) Limits in Modified Categories of Interest. Bulletin of the Iranian Mathematical Society, 43(7):2617-2634.
  • Fenn, R., C. Rourke (1992) Racks and links in codimension two. J. Knot Theory Ramifications, 1(4):343–406.
  • Martins, J.F. (2016) Crossed modules of Hopf algebras and of associative algebras and two-dimensional holonomy. J. Geom. Phys., 99:68–110.
  • Nelson, S. (2016) What is...a quandle?, Notices of AMS, 63:378–380.
  • Shammu, N. M. (1987) Algebraic and categorical structure of categories of crossed modules of algebras. Ph.D. Thesis. King’s College.
  • Whitehead, J. H. C. (1949) Combinatorial homotopy. i. Bull. Amer. Math. Soc., 55(3):213–245.
Year 2022, Volume: 4 Issue: 2, 56 - 68, 31.12.2022
https://doi.org/10.54286/ikjm.1199290

Abstract

References

  • Alp, M. (1998) Pullbacks of crossed modules and cat-1 groups. Turkish Journal of Mathematics, 22:273-281.
  • Alp, M. (2006) Pullbacks of crossed modules and cat-1 commutative algebras, Turkish Journal of Mathematics, 30:237-246.
  • Brown, R., Wensley, C.D. (1995) On infinite induced crossed modules and the homotopy 2-type of mapping cones. Theory Appl. Categ., 1.
  • Crans, A.S., Wagemann, F. (2014) Crossed modules of racks. Homology Homotopy Appl., 16(2):85–106.
  • Davvaz, B., Emir, K. (2020) On morphisms of crossed polymodules. Notices of AMS, 9(1):95-103.
  • Ege Arslan, U., Onarlı, G., (2015) An embedding theorem for the category of crossed p-modules. Georgian Mathematical Journal, 22(3):361–371.
  • Emir, K., Gülsün Akay, H. (2019) Pullback crossed modules in the category of racks. Hacettepe Journal of Mathematics and Statistics, 48(1):140-149.
  • Emir, K., Çetin, S. (2017) Limits in Modified Categories of Interest. Bulletin of the Iranian Mathematical Society, 43(7):2617-2634.
  • Fenn, R., C. Rourke (1992) Racks and links in codimension two. J. Knot Theory Ramifications, 1(4):343–406.
  • Martins, J.F. (2016) Crossed modules of Hopf algebras and of associative algebras and two-dimensional holonomy. J. Geom. Phys., 99:68–110.
  • Nelson, S. (2016) What is...a quandle?, Notices of AMS, 63:378–380.
  • Shammu, N. M. (1987) Algebraic and categorical structure of categories of crossed modules of algebras. Ph.D. Thesis. King’s College.
  • Whitehead, J. H. C. (1949) Combinatorial homotopy. i. Bull. Amer. Math. Soc., 55(3):213–245.
There are 13 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Hatice Gülsün Akay 0000-0001-7983-6852

İbrahim İlker Akça

Early Pub Date December 31, 2022
Publication Date December 31, 2022
Acceptance Date December 23, 2022
Published in Issue Year 2022 Volume: 4 Issue: 2

Cite

APA Gülsün Akay, H., & Akça, İ. İ. (2022). Completeness of the Category of Rack Crossed Modules. Ikonion Journal of Mathematics, 4(2), 56-68. https://doi.org/10.54286/ikjm.1199290
AMA Gülsün Akay H, Akça İİ. Completeness of the Category of Rack Crossed Modules. ikjm. December 2022;4(2):56-68. doi:10.54286/ikjm.1199290
Chicago Gülsün Akay, Hatice, and İbrahim İlker Akça. “Completeness of the Category of Rack Crossed Modules”. Ikonion Journal of Mathematics 4, no. 2 (December 2022): 56-68. https://doi.org/10.54286/ikjm.1199290.
EndNote Gülsün Akay H, Akça İİ (December 1, 2022) Completeness of the Category of Rack Crossed Modules. Ikonion Journal of Mathematics 4 2 56–68.
IEEE H. Gülsün Akay and İ. İ. Akça, “Completeness of the Category of Rack Crossed Modules”, ikjm, vol. 4, no. 2, pp. 56–68, 2022, doi: 10.54286/ikjm.1199290.
ISNAD Gülsün Akay, Hatice - Akça, İbrahim İlker. “Completeness of the Category of Rack Crossed Modules”. Ikonion Journal of Mathematics 4/2 (December 2022), 56-68. https://doi.org/10.54286/ikjm.1199290.
JAMA Gülsün Akay H, Akça İİ. Completeness of the Category of Rack Crossed Modules. ikjm. 2022;4:56–68.
MLA Gülsün Akay, Hatice and İbrahim İlker Akça. “Completeness of the Category of Rack Crossed Modules”. Ikonion Journal of Mathematics, vol. 4, no. 2, 2022, pp. 56-68, doi:10.54286/ikjm.1199290.
Vancouver Gülsün Akay H, Akça İİ. Completeness of the Category of Rack Crossed Modules. ikjm. 2022;4(2):56-68.

Cited By