Group-groupoid actions and liftings of crossed modules

Abstract

The aim of this paper is to define the notion of lifting of a crossed module via a group morphism and give some properties of this type of the lifting. Further we obtain a criterion for a crossed module to have a lifting of crossed module. We also prove that the liftings of a certain crossed module constitute a category; and that this category is equivalent to the category of covers of that crossed module and hence to the category of group-groupoid actions of the corresponding groupoid to that crossed module.

Liftings of crossed modules in the category of groups with operations

“…Mucuk and Şahan in [25] recently defined the notion of lifting for crossed modules in the category of groups and proved that the liftings of a certain crossed module are categorically equivalent to the actions of associated group-groupoid on groups.…”

“…For all a ∈ A and x ∈ X, we have Proof. Assume that ϕ : A → X is a crossed module in C. Since by [25,Theorem 4.2.] we have the following equality, ϕ((a, x) + (a ′ , x ′ )) = ϕ(a, x) + ϕ(a ′ , x ′ ) for a, a ′ ∈ A and x, x ′ ∈ X, we only need to prove that ϕ : A ⋊ X → X preserves the operations of Ω ′ 2 and Ω ′ 1 .…”

“…The object of this paper is to extend the results given in [25] to a more general certain category C. First we give the notion of lifting crossed module of a crossed module in C. Then we observe some properties of such lifting crossed modules. Finally we prove that for an internal groupoid G in C, internal groupoid actions of G on groups with operations and liftings of the crossed module associated with G are categorically equivalent.…”

“…On the other hand in [25], Mucuk and Şahan have recently defined the notion of lifting for crossed modules. If (A, B, α) is a crossed module and θ : X → B is a morphism of groups, then a crossed module (A, X, ϕ) in which the action of X on A is defined via θ such that θϕ = α is called a lifting of α over θ.…”

“…If (A, B, α) is a crossed module and θ : X → B is a morphism of groups, then a crossed module (A, X, ϕ) in which the action of X on A is defined via θ such that θϕ = α is called a lifting of α over θ. Also they proved in [25] that the liftings of a certain crossed module are categorically equivalent to the actions of associated group-groupoid on groups. See also [30] for further works on lifting crossed modules.…”