The notions of magnetic difference operator or magnetic exterior derivative defined on weighted graphs are discrete analogues of the notion of covariant derivative on sections of a fibre bundle and its extension on differential forms. In this paper, we extend these notions to certain 2-simplicial complexes called triangulations, in a manner compatible with changes of gauge. Then we study the magnetic Gauß-Bonnet operator naturally defined in this context and introduce the geometric hypothesis of χ−completeness which ensures the essential self-adjointness of this operator. This gives also the essential self-adjointness of the magnetic Laplacian on triangulations. Finally we introduce an hypothesis of bounded curvature for the magnetic potential which permits to caracterize the domain of the self-adjoint extension.