The term categorical trace usually refers to the concept introduced in
Nora Ganter and Mikhail Kapranov, Representation and character theory in 2-categories (arXiv, blog)
Let be a 2-category and a 1-endomorphism on one of its objects. Then the categorical trace of is the collection of 2-morphisms from the identity on into
Tr(F) = Hom_C(Id_B, F)
For , the 2-category of Kapranov-Voevodsky 2-vector spaces, a 1-endomorphism is an -matrix of vector spaces . The categorical trace on that is the direct sum of the diagonal entries, . Hence under the decategorification functor from to vectors and matrices with entries in , this becomes the ordinary trace of matrices.
The categorical trace is closely related to the span trace.
Created on January 29, 2009 21:53:38
by Urs Schreiber