Warning: "weakness" (no coproduct, no left adjoint of forget and others: viz no sets inside or too many) are "cured" in bounded (by 1) metrics. I'll write that, but it's better to present first Ban the Banach unitary balls with short linear maps. (till now, Ban is linked only from here).
Of course, there is the excelent, but unusual, category using "extended real number line and allow the distance function d to attain the value ∞."
the following wasn't accepted (?!) in metric space. It will be explained here:
A logical analysis
- The fundamental metric concept is short map, the morphisms of the metric category (isomorphisms, i.e. bishort maps, are the isometries), but its usual expresion uses order and addition in positive reals so,
- 1) It is obvious that : | x - |x - y | | = y is the same that x = 0 or y ≤ x, so distance in positive reals gives weak order there, strong order (y ≤ x iff ... ) is difficult, but possible, if we accept a solution of |x - y | = y i.e. y = x / 2.
- 2) | d(y, z) - |d(y, z) - d´(f(y), f(z)) | | = d´(f(y), f(z))
is the same to say that f is a short map, without any reference to order in positive reals.
- Really? Doesn't the absolute value function refer to order?? After all, it essentially detects whether a given real number is ≥ 0. Revolver 28 June 2005 04:15 (UTC)
- 3) The following equivalence of triangle inequality
- | d(x, y) - d(x, z) | ≤ d(y, z) says,
- Joined together : | d(y, z) - |d(y, z) - | d(x, y) - d(x, z) | | | = | d(x, y) - d(x, z) | is triangle inequality.
- slight change and : | d(y, z) - |d(z, y) - | d(x, y) - d(x, z) | | | = | d(x, y) - d(x, z) | is triangle inequality and simmetry (make z = x and use | x - d(y, y)| = x).
magma operation: XxX --> X
usual way metric: XxX --> R+ (wrong!)
But take a arbitrary but fixed form da: x |--> d(a,x) and add it to all bounded X --> R+ this will be R+X with the supreme metric. (The same that the bounded metric given by the supreme norm).
This gives metric as (isometric image) d: X --> R+X.
Now any dx is a short map : X --> R+ because this is exactly (equivalent to) triangle inequality, so visible in Met.
But R+ is metric so we have |.-.|: R+ --> R+R+ and |x-.|: R+ --> R+ as before (A logical analysis) but not medial: x - y is medial, x v y is medial but not (x-y) v (y-x)!
What if you want distance-preserving maps as morphisms? You need this category, e.g. when talking about metric completions, I believe.