Re: Non-cartesian closedness of Met

[Dirk Hofmann <dirk@ua.pt>]

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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