Re: symmetric monoidal traces

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

As Phil Scott immediately pointed out to me, the proof of this
folklore result appears in a 2000 slide by Masahito Hasegawa entitled
"A short proof of the uniqueness of trace on tortile categories", see

 http://www.kurims.kyoto-u.ac.jp/~hassei/papers/canonicaltrace.gif

and also on p.23 in a paper by the same author that is to appear in
MSCS, entitled "On traced monoidal closed categories":

 http://www.kurims.kyoto-u.ac.jp/~hassei/papers/tmcc-revised16may08.pdf

-- Peter


Peter Selinger wrote:
>
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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
>
>