Crossed modules bifibred over k-Algebras

Abstract

In this paper we examine on a pair of adjoint functors (𝜙∗ , 𝜙∗)for a subcategory of the category of crossed modules over commutative algebras where 𝜙∗: XMod/𝑃 → XMod/ Q, induced, and 𝜙∗: XMod/Q → XMod/𝑃, pullback (co-induced), which enables us to move from crossed 𝑄-modules to crossed P-modules by an algebra morphism 𝜙 : P → Q. We show that this adjoint functor pair (𝜙∗, 𝜙∗) makes 𝑝 ∶ XMod → k-Alg into a bi- fibred category over k-Alg, the category of commutative algebras, where p is given by 𝑝(𝐶, 𝑅, 𝜕) = 𝑅. Also, we give some examples and results on induced crossed modules in the case when 𝜙 is an epimorphism or the inclusion of an ideal.