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](https://arxiv.org/abs/1107.6032v2) 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 and what is special about the trace that makes it amenable to categorification in such generality?



Categorification of determinant
There is a notion of determinant functor, they were introduced for abelian and exact categories by P. Deligne in his paper "Le déterminant de la cohomologie" (<https://publications.ias.edu/sites/default/files/Number58.pdf>).
There is an extension to categories of bounded complexes by F. Knudsen and D. Mumford.


More recently you also have versions for triangulated categories, cf this paper by M. Breuning "Determinant functors on triangulated categories" and also by F. Muro, A. Tonks and M. Witte "On determinant functors and K-theory".


