Right Trace (Traced Monoidal Category)

In a Traced Monoidal Category $𝓒$, a right trace is a Natural Transformation

$$\mathrm{t}{\mathrm{r}}_{\mathrm{R}}:\prod _{X:𝓒}𝓒[A\otimes X,B\otimes X]\Rightarrow 𝓒[A,B]\forall A,B:𝓒$$

Written with the first parameter as superscript:

$${\mathrm{t}{\mathrm{r}}_{\mathrm{R}}}^X:𝓒[A\otimes X,B\otimes X]\Rightarrow 𝓒[A,B]\forall A,B,X:𝓒$$

It can be thought of as a feedback loop from the output back into the input. of a Functor.

Axioms

A right trace must satisfy the following axioms.

Given $A,B,C,D,X:𝓒$

Tightening

$$\begin{array}{rlrl}& \text{Given }& & h:A\to B,\\ & & & f:B\otimes X\to C\otimes X,\\ & & & g:C\to D\end{array}$$ $$\begin{array}{r}{\mathrm{tr}}_R^X((g\otimes 1_X)\circ f\circ (h\otimes 1_X))=g\circ {\mathrm{tr}}_R^X(f)\circ h\end{array}$$

This states that if a trace passes through an arrow that does’t interact with the trace, then the scope of the trace can be contracted.

Sliding

$$\begin{array}{rlrl}& \text{Given }& & f:A\otimes D\to B\otimes C,\\ & & & g:C\to D\end{array}$$ $$\begin{array}{r}{\mathrm{tr}}_R^X((1_B\otimes g)\circ f)={\mathrm{tr}}_R^X(f\circ (1_A\otimes g))\end{array}$$

This ammounts to sliding a function around a functor $g$ around the trace so it ends up occurring before $f$ rather than after.

Vanishing

$$\begin{array}{rl}\text{Given }& f:A\otimes (C\otimes D)\to B\otimes (C\otimes D),\\ & g:A\otimes 1\to B\otimes 1,\end{array}$$ $$\begin{array}{r}{\mathrm{tr}}_R^{C\otimes D}(f)={\mathrm{tr}}_R^C({\mathrm{tr}}_R^D(f))\\ {\mathrm{tr}}_R^1(f)=f\end{array}$$

The first expresses a kindof currying of traces and the second states that a trace functor over the monoidal identity is the identity.

Strength

$$\begin{array}{rl}\text{Given }& f:A\otimes X\to B\otimes X\\ & g:C\to D,\end{array}$$ $$\begin{array}{r}{\mathrm{tr}}_R^X(g\otimes f)=g\otimes {\mathrm{tr}}_R^X(f)\end{array}$$

This is similar to tightening but in the other dimension. It expresses that arrows which don’t interact with trace, can be taken out of the scope of the trace.