nLab
categorical trace

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)

Definition

Let CC be a 2-category and F:BBF : B \to B a 1-endomorphism on one of its objects. Then the categorical trace Tr(F)Tr(F) of FF is the collection of 2-morphisms from the identity Id BId_B on BB into BB

Tr(F)=Hom C(Id B,F). Tr(F) = Hom_C(Id_B, F) \,.

Examples

  • For C=KV2VectC = KV2Vect, the 2-category of Kapranov-Voevodsky 2-vector spaces, a 1-endomorphism Vect nFVect nVect^n \stackrel{F}{\to} Vect^n is an n×nn \times n-matrix (V ij) i,j(V_{i j})_{i,j} of vector spaces V ijVectV_{i j} \in Vect. The categorical trace on that is the direct sum of the diagonal entries, Tr(F)=oplu iV iiTr(F) = \oplu_{i} V_{i i}. Hence under the decategorification functor from KV2VectKV2Vect to vectors and matrices with entries in \mathbb{N}, this becomes the ordinary trace of matrices.

Remarks

  • The categorical trace is closely related to the span trace.
Created on January 29, 2009 21:53:38 by Urs Schreiber (134.100.222.156)