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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