nLab
trace

Definition

If $a$ is a dualizable object in a symmetric monoidal category $C$, there is a notion of the trace of an endomorphism $f:a\to a$, which reproduces the ordinary notion of trace of a linear map of finite dimensional vector spaces for the case that $C=\mathrm{Vect}$.

The idea of the trace operation is easily seen in string diagram notation: essentially one takes the endomorphism $a\stackrel{f}{\to }a$, “bends it around” using the duality and the symmetry and connects its output to its input.

$$\begin{array}{c}1\\ {\downarrow }^{\mathrm{tr}(f)}\\ 1\end{array}:=\begin{array}{cc}& 1\\ & \downarrow \\ a^{*}& \otimes & a\\ {\downarrow }^{{\mathrm{Id}}_{a^{*}}}& & {\downarrow }^f\\ a^{*}& \otimes & a\\ & {\downarrow }^{b_{a^{*},a}}\\ a& \otimes & a^{*}\\ & \downarrow \\ & 1\end{array}$$\array{ 1 \\ \;\;\;\downarrow^{tr(f)} \\ 1 } \;\;\; := \;\;\; \array{ & 1 \\ & \downarrow \\ a^* &\otimes& a \\ \;\;\;\downarrow^{Id_{a^*}} && \;\;\downarrow^f \\ a^* &\otimes& a \\ & \downarrow^{b_{a^*, a}} \\ a &\otimes& a^* \\ & \downarrow \\ & 1 }

This definition makes sense in any braided monoidal category, but often in non-symmetric cases one wants instead a slightly modified version which requires the extra structure of a balancing?.

The trace of the identity $1_a:a\to a$ is called the dimension or Euler characteristic of $a$.

Examples

• $C=\mathrm{Vect}$ with its standard monoidal structure (tensor product of vector spaces): in this case tr(f) is the usual trace of a linear map;

• $C=\mathrm{SuperVect}=({\mathrm{Vect}}_{{\mathbb{Z}}_2},\otimes ,b)$, the category of ${\mathbb{Z}}_2$-graded vector spaces with the nontrivial symmetric braiding which is $-1$ on two odd graded vector spaces: in this case the above is the supertrace on supervectorspaces, $\mathrm{str}(V)=\mathrm{tr}(V_{\mathrm{even}})-\mathrm{tr}(V_{\mathrm{odd}})$.

• $C=\mathrm{Span}({\mathrm{Top}}^{\mathrm{op}})$: here the trace is the co-span co-trace which can be seen as describing the gluing of in/out boundaries of cobordisms

• $C=\mathrm{Span}(\mathrm{Grpd})$: this reproduces the notion of trace of a linear map within the interpretation of spans of groupoids as linear maps in the context of groupoidification and geometric function theory, made explicit at span trace

Categorification

See trace of a category

References

• Joyal, Street, and Verity, Traced Monoidal Categories
• Dold, Albrecht and Puppe, Dieter, Duality, trace, and transfer
• Peter Selinger, A survey of graphical languages for monoidal categories (pdf), Section 5