The notion of trace of a matrix can be generalized to trace of an endomorphism of a dualizable objects in a symmetric monoidal category. (See Ponto & Shulman for a nice description.)

Is there a categorification of the notion of determinant as well? If it exists, where can I read about it? If it doesn't exist, what is the conceptual obstruction to it, or what is special about the trace that makes it amenable to categorification in such generality?