SAMS Subject Classification Number: 4

Internal categorical structures play an important role in categorical algebra. In semi-abelian categories, internal categories form a variety, namely the variety of crossed modules [1]. The notion of star multiplication was introduced by G. Janelidze in [1] where it was applied to the description of crossed modules in a semi-abelian category. In the same paper, the question of describing semi-abelian categories with the property that every star-multiplicative graph uniquely extends to an internal category structure was asked.

N. Martins-Ferreira [3] introduced conditions that provide a simple description of internal groupoids as crossed modules in the semi-abelian categories of groups and rings. The aim of this talk is to describe star-multiplication in varieties of right \(\Omega\)-loops in the sense of [2].

[1] G. Janelidze, Internal crossed modules, Georgian Math.J. 10, 1 (2003), 99-114.

[2] E. B. Inyangala, Semidirect products and crossed modules in varieties of right \(\Omega\)-loops, Theory Appl. Categories 25, 16 (2011), 426-435.

[3] N. Martins-Fereira, Star-multiplication in pointed protomodular categories, Theory Appl. categories 23, 9 (2010), 170-198.