Non-cartesian closedness of Met

Non-cartesian closedness of Met

[ptj]

Let Met denote the category of metric spaces and nonexpansive maps.
It's well known that if we equip the product of two metric spaces
with the L_{\infty} metric (the max of the distances in the two
coordinates), we get categorical products in Met; alternatively,
if we impose the L_1 metric on the product (the sum of the two
coordinate distances), we get a monoidal closed structure, at least
if we weaken the usual definition of a metric by allowing metrics to
take the value \infty.

It's intuitively obvious that the cartesian monoidal structure on Met
can't be closed. But I've never (until I wrote one down today!) seen
a formal proof of this; does anyone know if it exists anywhere in the
literature? My proof is not particularly elegant: it amounts to showing
that a particular coequalizer in Met is not preserved by a functor of
the form (-) x Y.



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Re: Non-cartesian closedness of Met (ptj@maths.cam.ac.uk)

[Vaughan Pratt]

Peter, can you generalize your argument to a necessary and sufficient
condition on V such that the usual closed structure on V-Cat is cartesian?

This is of interest for its applicability to models of concurrent
computation, as follows.

My group's 1989 paper "Temporal Structures",

http://boole.stanford.edu/pub/man90.pdf

presented at CTCS-89 in Manchester and later in journal form as  Mathematical
Structures in Computer Science, Volume 1:2, 179-213 (July 1991), had the
following abstract.

"We combine the principles of the Floyd-Warshall-Kleene algorithm, enriched
categories, and Birkhoff arithmetic, to yield a useful class of algebras of
transitive vertex-labeled spaces.  The motivating application is a uniform
theory of abstract or parametrized time in which to any given notion of
time there corresponds an algebra of concurrent behaviors and their
operations, always the same operations but interpreted automatically and
appropriately for that notion of time.  An interesting side application is
a language for succinctly naming a wide range of datatypes."

A little later I noticed that the structures treated there were starting to
look like those in Girard's recent (1986) work on linear logic, and wrote
about that as Section 5 in my CONCUR-92 paper "The Duality of Time and
Information",  http://boole.stanford.edu/pub/dti.pdf.  (I like to think of
Ecclesiastes 9:11, "but time and chance happeneth to them all" as an early
appearance of that duality via Shannon's statistical view of information,
and the bras and kets of quantum mechanics as a later one.)

The idea of orthocurrence (the monoid in a closed monoidal category) as
interaction, and its adjoining closed structure as observation, is spelled
out more explicitly in a talk I gave at an IJCAI'01 workshop on spatial and
temporal reasoning in Seattle in 2001 organized by NSF's Frank Anger.

Slides (cryptic): http://boole.stanford.edu/pub/ijcaitalk.pdf
Paper (detailed): http://boole.stanford.edu/pub/ortho.pdf, unpublished but
later incorporated into
Paper (far more detailed): http://boole.stanford.edu/pub/seqconc.pdf

What got me into all this in the beginning was noticing in the mid-1980s
that in the obvious generalization of ordered time (Pos) to real time
(Met), the closed monoidal structure ceased to be cartesian and the single
operation of orthocurrence now split into two monoids, one closed and the
other cartesian.

That's when I noticed the similarity to Girard's linear logic, which had
made the same split independently and at the same time but for a totally
different application, substructural logic.  The connection was completed
once I'd figured out the duality of time and information as in Ezekiel 9:11.

For Vineet Gupta's thesis in 1991 I suggested choosing between that duality
and cubical complexes as a geometric model of concurrency.  After looking
at both for a month Vineet picked the former.  At POPL'91 Boris
Trakhtenbrot asked me at question time how the two could be connected,
which I was unable to do until realizing (too late for Vineet who'd
finished his thesis by then) that Chu(Set,3) provided the edges, faces,
etc. of the cubical complexes by interpreting 3 as {0, 1/2, 1} with each
face's dimension given by the number of 1/2's (transitions) appearing in it.

Vaughan Pratt


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Re: Non-cartesian closedness of Met

[Jirí Adámek]

Dear Peter,

Thanks for pointing out that my argument has been incomplete. Here
is a proof that Met is not a ccc:

We use that limits in Met are the limits in Set with the supremum
metric (coordinate-wise). Let us denote by Met(X,Y) the hom-sets with the
supremum metric d'(f,g) = sup_x d(fx,gx). The addition of real
numbers from RxR to R is not non-expanding, but its curried from
R to Met(R,R) is; thus all we need to do is to show that if Met were a
ccc, one could take [X,Y]=Met(X,Y).

The underlying set of [X,Y] can be taken to be the hom-set, using
adjoint transposes of morphisms from 1 to [X,Y]. And the universal
morphism eval:[X,Y]xX -> Y can be taken to be the evaluation map
(precompose it with morphisms fxX for f:1->[X,Y]). The metric d of
[X,Y] satisfies d \geq d', using adjoint transposes of morphisms
from 2-element spaces to [X,Y]. To prove d \leq d', we first consider
a finite space X. Let id: n->X be the identity map from the discrete
space on the underlying set of X, a coproduct of n copies of 1. Then
[id,Y]: [X,Y]->[n,Y]= Y^n demonstrates that d= d'. For X arbitrary,
express it as a directed colimit X=colim X_i of all finite subspaces X_i.
Then [X,Y] = lim [X_i,Y] carries the supremum metric.

Best regards,
Jiri


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Re: Non-cartesian closedness of Met

[ptj]

Dear Jirka,

That's an interesting idea, but I'm having difficulty making it work
in practice. It seems to me that the most you can get from considering
morphisms from 2 is that the metric on [X,Y] must satisfy
d(f,g) \geq sup_x d(fx,gx) -- to get the reverse inequality you would 
need to impose the L_1 metric on the product 2 x X. And that
inequality is the wrong way round for showing that the transpose
of addition on R is nonexpansive.

Best regards,
Peter

On Dec 16 2022, Jirí Adámek wrote:

>Hi Peter,
>
>If Met is a CCC, then [X,Y] has as elements all morphisms from X to Y
>(use the adjoint transposes of morphisms from 1 to [X,Y]). And the 
>distance of morphisms f,g is sup_x d(fx,gx) (use the adjoint transposes
>of morphisms from two-element spaces to [X,Y]).
>
>However, addition of real numbers is not nonexpansive from R x R to R, 
>although its curred form from R to [R,R] is. This is a contradiciotn.
>
>Best regards,
>Jiri
>
>On Fri, 16 Dec 2022, ptj@maths.cam.ac.uk wrote:
>
>> Let Met denote the category of metric spaces and nonexpansive maps.
>> It's well known that if we equip the product of two metric spaces
>> with the L_{\infty} metric (the max of the distances in the two
>> coordinates), we get categorical products in Met; alternatively,
>> if we impose the L_1 metric on the product (the sum of the two
>> coordinate distances), we get a monoidal closed structure, at least
>> if we weaken the usual definition of a metric by allowing metrics to
>> take the value \infty.
>>
>> It's intuitively obvious that the cartesian monoidal structure on Met
>> can't be closed. But I've never (until I wrote one down today!) seen
>> a formal proof of this; does anyone know if it exists anywhere in the
>> literature? My proof is not particularly elegant: it amounts to showing
>> that a particular coequalizer in Met is not preserved by a functor of
>> the form (-) x Y.
>>
>>


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Re: Non-cartesian closedness of Met

[Dirk Hofmann]

Dear Peter,

we don't have such an example in the paper. Our argument used the 
cartesian closed category of "metric spaces without triangular 
inequality", and showed that, for a metric space X, all 
exponential Y^X (Y metric) taken in that category satisfy the 
triangular inequality if and only if X satisfies that condition.

Best regards
Dirk




On 16 December 2022 at 22:05 GMT+0000, ptj@maths.cam.ac.uk wrote 
...
> Dear Dirk, dear Maria Manuel,
>
> That's very interesting, and I should have remembered it. But 
> did
> your argument come up with an explicit example of a colimit in 
> Met
> not preserved by a functor of the form (-) x Y ?
>
> I suppose I'll have to go and read your paper!
>
> Best regards,
> Peter
>
> On Dec 16 2022, Dirk Hofmann wrote:
>
>>
>>Dear Peter,
>>
>>in our paper
>>
>> - Clementino, M. M., & Hofmann, D. (2006). Exponentiation in
>> $V$-categories. Topology and its Applications, 153(16), 
>> 3113–3128.
>>
>> we give a characterisation of exponentiable metric spaces. The 
>> result
>> essentially states that a metric space (in the sense of 
>> Lawvere) is
>> exponentiable if and only if "there is always a point in the 
>> middle",
>> that is, whenever d(x,z)=u+v, then there is a point y with 
>> d(x,y)≤u+ε
>> and d(y,z)≤v+ε. A finite metric space with a non-trivial 
>> distance
>> cannot be exponentiable.
>>
>>Best regards
>>Dirk
>>
>> On 16 December 2022 at 16:41 GMT+0000, <ptj@maths.cam.ac.uk> 
>> wrote ...
>>> Let Met denote the category of metric spaces and nonexpansive 
>>> maps.
>>> It's well known that if we equip the product of two metric 
>>> spaces
>>> with the L_{\infty} metric (the max of the distances in the 
>>> two
>>> coordinates), we get categorical products in Met; 
>>> alternatively,
>>> if we impose the L_1 metric on the product (the sum of the two
>>> coordinate distances), we get a monoidal closed structure, at 
>>> least
>>> if we weaken the usual definition of a metric by allowing 
>>> metrics to
>>> take the value \infty.
>>>
>>> It's intuitively obvious that the cartesian monoidal structure 
>>> on Met
>>> can't be closed. But I've never (until I wrote one down 
>>> today!) seen
>>> a formal proof of this; does anyone know if it exists anywhere 
>>> in the
>>> literature? My proof is not particularly elegant: it amounts 
>>> to
>>> showing
>>> that a particular coequalizer in Met is not preserved by a 
>>> functor of
>>> the form (-) x Y.
>>>
>>>


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