Cleft extensions and Galois extensions for Hom-associative algebras (Q2797909)

Different types of Hom-associative algebra have been considered in the literature. In present paper a unital Hom-associative algebra in a strict monoidal category \((\mathcal{C},\otimes,K)\) is defined as an object \(A\) together with morphisms \(\mu:A\otimes A\to A\), \(\eta:K\to A\) and \(\alpha:A\to A\) subject to the equalities \(\mu\circ(A\otimes\eta)=id_A=\mu\circ(\eta\otimes A)\) and \(\mu\circ(\alpha\otimes\mu)=\mu\circ(\mu\otimes\alpha)\) (note that \(\alpha\) is not assumed to me multiplicative). The dual notion of counital Hom-coassociative coalgebra is also considered. Under mild assumptions on \(\mathcal{C},\) the machinery of cleft extensions and Galois extensions is developed in this setting obtaining the appropriate analogue of the classical results.