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 $\mathrm{Tr}(F)$ of $F$ is the collection of 2-morphisms from the identity ${\mathrm{Id}}_B$ on $B$ into $B$

$$\mathrm{Tr}(F)={\mathrm{Hom}}_C({\mathrm{Id}}_B,F).$$Tr(F) = Hom_C(Id_B, F) \,.

Examples

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