Completeness of the Category of Rack Crossed Modules

Year 2022, Volume: 4 Issue: 2, 56 - 68, 31.12.2022

Hatice Gülsün Akay İbrahim İlker Akça

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} $.

Keywords

Crossed module, limit, 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.