The category of generalized crossed modules

Document Type: Research Paper


Department of Mathematics, Shahid Beheshti University, G.C., Tehran 19839, Iran


In the definition of a crossed module $(T,G,\rho)$, the actions of the group $T$ and $G$ on themselves are given by conjugation. In this paper, we consider these actions to be arbitrary and thus generalize the concept of ordinary crossed module. Therefore, we get the category ${\bf GCM}$, of all generalized crossed modules and generalized crossed module morphisms between them, and investigate some of its categorical properties. In particular, we study the relations between epimorphisms and the surjective morphisms, and thus generalize the corresponding results of the category of (ordinary) crossed modules. By generalizing the conjugation action, we can find out what is the superiority of the conjugation to other actions. Also, we can find out a generalized crossed module with which other actions (other than the conjugation) has the properties same as a crossed module.


[1] Adamek, J., Herrlich, H., and Strecker, G.E., "Abstract and Concrete Categories", John Wiley and Sons, 1990.
[2] Alp, M., Pullbacks of crossed modules and cat1-groups, Turk. J. Math. 22 (1998),273-281.
[3] Brown, B., Coproducts of crossed P-modules: Applications to second homotopy groups and to the homology of groups, Topology 23(3) (1984), 337-345.
[4] Loday, J.L., Spaces with finitely many non-trivial homotopy groups, J. Pure Appl. Algebra 24 (1982), 179-202.
[5] Mac Lane, S. and Whitehead, J.H.C., On the 3-type of a complex, Proc. Nat. Acad. Sci. 36(1) (1950), 41-48.
[6] Mohammadzadeh, H., Shahrokhi, S., and Salemkar, A.R. Some results on stem covers of crossed modules, J. Pure Appl. Algebra 218 (2014), 1964-1972.
[7] Norrie, K., "Crossed modules and analogues of group theorems", Dissertation, King's College, University of London, 1987.
[8] Rodeja, E.G., Ladra, M., and Lopez, M.P., Epimorphisms in the category of crossed modules, Southeast Asian Bull. Math. 28 (2004), 287-302.
[9] Salemkar, A.R., Mohammadzadeh, H., and Shahrokhi, S., Isoclinism of crossed mod