Transformation groupoids associated to group actions capture the interplay between global and local symmetries of structures described in set-theoretic terms. This paper examines the analogous situation for structures described in category-theoretic terms, where symmetry is expressed as the action of a 2-group G (equivalently, a categorical group) on a category C. It describes the construction of a transformation groupoid in diagrammatic terms, and considers this construction internal to Cat, the category of categories. The result is a double category C//G which describes the local symmetries of C. We define this and describe some of its structure, with the adjoint action of G on itself as a guiding example.
Keywords: 2-group, categorical group, crossed module, action, double category, adjoint action
2010 MSC: 18B40, 18D10, 20L99
Theory and Applications of Categories, Vol. 30, 2015, No. 43, pp 1429-1468.