Non-cartesian closedness of Met

[ptj@maths.cam.ac.uk]

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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