Cat¹-groups and crossed modules are equivalent formulations of 2-groups (a two-dimensional generalisation of the concept of group). A linear representation of the cat¹-group is defined to be a 2-functor chain complexes of vector spaces over a fixed field. This definition, and to a large extent the theory of cat¹-group representations, is based on analogy with the classical theory of group representations. There exist cat¹ analogues of many definitions and results from
group representation theory. These include the notion of a faithful representation and the existence of regular representations given by Cayley's theorem. However, there are also divergences between the theories. For example, the regular representation for cat¹-groups is not necessarily faithful. Every cat¹-group, has an associated cat¹-group algebra I<, which is obtained by first applying the group algebra functor to and then factoring the top level by a given ideal in order to introduce relations necessary to make the kernel conditions work in the algebra. Representations of a cat¹-group are equivalent to modules over its cat¹-group algebra. Since representations are 2-functors,
there is a 2-functor 2-category Repc of representations, morphisms between them (natural transformations), and homotopies between the morphisms (modifications). Many of the results on the structure of group representations, for example Maschke's theorem, will generalise to the next dimension, although we have only just begun to scratch the surface of this theory. Since it is possible to pass freely between cat¹-groups and crossed modules, it is also possible to describe representation theory for 2-groups in the language of crossed modules.