Representation and character theory in 2-categories

Abstract

We develop a (2-)categorical generalization of the theory of group representations and characters. We categorify the concept of the trace of a linear transformation, associating to any endofunctor of any small category a set called its categorical trace. In a linear situation, the categorical trace is a vector space and we associate to any two commuting self-equivalences a number called their joint trace. For a group acting on a linear category V we define an analog of the character which is the function on commuting pairs of group elements given by the joint traces of the corresponding functors. We call this function the 2-character of V. Such functions of commuting pairs (and more generall..