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 $C$ be a 2-category and $F : B \to B$ a 1-endomorphism on one of its objects. Then the categorical trace $Tr(F)$ of $F$ is the collection of 2-morphisms from the identity $Id_B$ on $B$ into $B$

$$Tr(F) = Hom_C(Id_B, F) \,.$$

Examples

• For $C = KV2Vect$, the 2-category of Kapranov-Voevodsky 2-vector spaces, a 1-endomorphism $Vect^n \stackrel{F}{\to} Vect^n$ is an $n \times n$-matrix $(V_{i j})_{i,j}$ of vector spaces $V_{i j} \in Vect$. The categorical trace on that is the direct sum of the diagonal entries, $Tr(F) = \oplu_{i} V_{i i}$. Hence under the decategorification functor from $KV2Vect$ 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.