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.