We develop a theory of twisted actions of categorical groups using a notion of semidirect product of categories. We work through numerous examples to demonstrate the power of these notions. Turning to representations, which are actions that respect vector space structures, we establish an analog of Schur's lemma in this context. Keeping new terminology to a minimum, we concentrate on examples exploring the essential new notions introduced.
Keywords: Representations; Double Categories; Categorical Groups; 2-Groups; Semidirect Products
2010 MSC: Primary18D05; Secondary: 20C99
Theory and Applications of Categories, Vol. 29, 2014, No. 8, pp 215-255.