SAMS Subject Classification Number: 4, 29

Given a semigroup \(S\) and a unital commutative ring \(K\), by an \(S\)-graded \(K\)-algebra we mean an \(S\)-indexed family \(A=(A_s)_{s\in S}\) of \(K\)-modules equipped with a family of \(K\)-bilinear multiplications \[(A_s \times A_t \rightarrow A_{st})_{s,t\in S}\qquad{(1)}\] all written as \((x,y) \mapsto xy\) and satisfying the associative condition \(x(yz)=(xy)z\) for all \(s,t,u\in S\) and \(x \in A_s\), \(y\in A_t\), \(z\in A_u\). A morphism \(f:A\rightarrow B\) of \(S\)-graded \(K\)-algebras is a family \(f=(f_s)_{s\in S}\) of \(K\)-module homomorphisms \(f_s:A_s \rightarrow B_s\) with \(f_s(x)f_t(y)=f_{st}(xy)\) for all \(s,t\in S\) and \(x\in A_s\), \(y\in A_t\). We denote the category of \(S\)-graded \(K\)-algebras by \(\textbf{Alg}(K,S)\).

In this talk we study various topics on the semi-abelian (in the sense of [1]) category of \(S\)-graded \(K\)-algebras, that is, Smith and Huq commutators, theory of split extensions, internal categories and crossed modules in the context of \(\textbf{Alg}(K,S)\). Smith commutator was first introduced for congruences of algebras in Mal’cev varieties in [2] and Huq commutator of pair of morphisms with same codomain in [3]. The notion of crossed modules was introduced in [4] in study of groups with the algebraic properties of relative homotopy groups.

[1] G. Janelidze, L. Márki, and W. Tholen, Semi-abelian categories, J. Pure Appl. Algebra 168 (2-3) (2002), 367–386.

[2] J.D.H Smith, Mal’cev varieties, Springer L.N. in Math 554 (1976).

[3] S.A Huq, Commutator, nilpotency, and solvability in categories, Quart. J. Oxford 19 (1968), 363–389.

[4] J.H.C Whitehead, Combinatorial homotopy II, Bull. Amer. Math. Soc. 55 (1949), 453–496.