Re: symmetric monoidal traces

[selinger@mathstat.dal.ca (Peter Selinger)]

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Dear Mike,

I don't know whether the proof of this result is spelled out in the
literature (uniqueness of the trace for dualizable objects in a
symmetric monoidal category). However, I have seen the result itself
mentioned, and it follows straightforwardly from Joyal-Street-Verity's
INT construction.

As you have already said, every traced symmetric monoidal category C
can be embedded in a compact closed category INT(C), in such a way
that the trace of C is mapped to the canonical trace of
INT(C). Further, every strong monoidal functor preserves dual objects,
so if some object X of C has a "canonical" trace coming from a dual
object, then this also gets mapped to the canonical trace of INT(C).
Finally, since the functor is faithful, and maps the "given" and the
"canonical" trace to the same thing, it follows that the two traces
already coincide in C. The same argument works for balanced monoidal
categories.

One can easily turn this argument into an elementary algebraic
proof. For an object X equipped with two traces Tr and Tr', consider
the following "interchange property" for f: A*X*X -> B*X*X:

Tr'_X(Tr_X(f o (A*c))) = Tr_X(Tr'_X((B*c) o f))

See Figure (a) in the attached file for an illustration of this
property. It is akin to "symmetry sliding", except that it uses two
different traces.

Figure (b) proves that if two traces satisfy the interchange property,
then they coincide.

Finally, if one of the traces is the canonical one obtained from a
dual of X, then the interchange property holds by standard
diagrammatic reasoning (see Figure (c)). In particular, if a dual
exists, then the trace on X is unique.

The proof, as shown in the attachment, is only correct in the
symmetric case. It also works in the balanced case, provided that one
inserts a twist map in the correct places.

-- Peter

(Attachment also available as
http://www.mathstat.dal.ca/~selinger/downloads/traces.gif)

Michael Shulman wrote:
>
> Hi all,
>
> Can someone please point me to whatever categorical references exist
> regarding the canonical trace for dualizable objects in a symmetric
> monoidal category?  I am particularly interested in (1) elementary
> expositions accessible to non-category-theorists and (2) any proofs of
> its uniqueness subject to various conditions.
>
> I know there are many references on traced monoidal categories,
> particularly with applications to computer science, but right now I am
> only interested in the symmetric monoidal trace.  I also know that
> there are various reinventions/expositions of the notion in, for
> example, the topological literature (e.g. Dold-Puppe), but I would
> like an exposition not tied to any particular application.  Finally, I
> know that the Joyal-Street-Verity paper "Traced Monoidal Categories"
> proves that the canonical symmetric (or, more precisely, balanced)
> monoidal trace is "universal" in that any traced monoidal category can
> be embedded in one equipped with the canonical trace, but as far as I
> can tell this need not determine the canonical trace uniquely.
>
> Thanks!!
> Mike