symmetric monoidal traces

symmetric monoidal traces

[Michael Shulman]

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

Re: symmetric monoidal traces

[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:
>
> [ Reminder from moderator: Attachments are not suitable for transmission
> and the one mentioned below has been deleted (happily Peter provides a
> url). On a similar note, recall that html messages are not suitable
> either, and will not be posted. Please send text only. Thanks. ]
>
> 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
>
>

Re: symmetric monoidal traces

[Peter Selinger]

[ Reminder from moderator: Attachments are not suitable for transmission
and the one mentioned below has been deleted (happily Peter provides a
url). On a similar note, recall that html messages are not suitable
either, and will not be posted. Please send text only. Thanks. ]

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