Group-groupoid actions and liftings of crossed modules

What is it about?

The aim of this paper is to define the notion of lifting via a group morphism for a crossed module and give some properties of this type of liftings. Further,we obtain a criterion for a crossed module to have a lifting crossed module. We also prove that the category of the lifting crossed modules of a certain crossed module is equivalent to the category of group-groupoid actions on groups, where the group-groupoid corresponds to the crossed module.

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Why is it important?

The theory of covering groupoids plays an important role in the applications of groupoids. In this theory, it is well known that for a groupoid G, the category of groupoid actions of G on sets, which are also called operations or G-sets, is equivalent to the category of covering groupoids of G.