Bol. Soc. Paran. Mat. (3s.) v. 38 7 (2020): 181–193. c SPM –ISSN-2175-1188 on line ISSN-00378712 in press SPM: www.spm.uem.br/bspm doi:10.5269/bspm.v38i7.46500 Liftings of Crossed Modules in the Category of Groups with Operations H. Fulya Akız, Osman Mucuk and Tun¸ car S ¸ahan abstract: In this paper we define the notion of lifting of a crossed module via the morphism in groups with operations and give some properties of this type of liftings. Further we prove that the lifting crossed modules of a certain crossed module are categorically equivalent to the internal groupoid actions on groups with operations, where the internal groupoid corresponds to the crossed module. Key Words: Internal groupoid, Covering groupoid, Group with operations, Lifting crossed module. Contents 1 Introduction 181 2 Preliminaries 182 3 Crossed modules in groups with operations 184 4 Liftings of crossed modules in groups with operations 187 5 Equivalences of the categories 191 1. Introduction Groupoids are mathematical structures which are known to be useful in many areas of science [5,16]. A groupoid is a small category in which each arrow has an inverse and a group-groupoid is an internal groupoid in the category of groups. For a group-groupoid G the covers of G in the category of group-groupoids are categor- ically equivalent to the group-groupoid actions of G on groups [7, Proposition 3.1]. A crossed module defined by Whitehead in [32,33] can be viewed as a 2-dimensional group [8] and has been widely used in homotopy theory [4], the theory of identities among relations for group presentations [9], algebraic K-theory [19], and homo- logical algebra [18,20]. See [4] for a discussion of the relation of crossed modules to crossed squares and so to homotopy 3-types. We refer the readers to [6] for the structure of the actor 2-crossed module related to the automorphisms of a crossed module of groupoids. In [10, Theorem 1] Brown and Spencer proved that group-groupoids are cate- gorically equivalent to crossed modules. Then in [29, Section 3], Porter proved that a similar result holds for a certain algebraic category C introduced by Orzech [27] 2010 Mathematics Subject Classification: 20L05, 57M10, 22A22. Submitted February 05, 2019. Published July 24, 2019 181 Typeset by B S P M style. c  Soc. Paran. de Mat.