[FRIAM] A/R theory

[jon zingale]

Unless I am somehow forgetting some clever interpretation, I was wrong about
the transitivity.

Let me try to reason from an example: an experimenter defines a litany of
tests for deciding how well a collection of things can be relied upon when
treated as computational objects. For instance, an audiophile may have a box
of capacitors that they wish to rank according to how well the caps filter
out hum without suppressing the dynamic range of the music. This process
defines a partition function on the box of capacitors. In a limiting case,
we can imagine having only two buckets, one with caps that are good enough
and the other with those that are not. In this coarse way, transitivity
holds because we either grabbed 3 caps that are from the *good enough*
bucket or we did not.

What I think I found confusing has to do with the distance function d:: C_t
x R_t(H) -> K, with K some ring. Here, allowing the C_t param to vary has
the effect of allowing the problem dependence to vary, or as in the example
above, allowing the hum tolerance to vary. Fixing a problem domain fixes the
C_T and this is rather instead like providing a space equipped with a fixed
origin. From that the more familiar distance function d':: R_T(H) x R_T(H)
-> K can easily be formed with nice transitivity features and all.

Now that I am reoriented a bit, I think an interpretation in terms of
V-profunctors and the closed monoidal categories we discussed in the linear
logic discussions could be fruitful. In effect, the function d as defined in
the paper is effectively a profunctor interpreted via a Cost quantale,
covariant in the Abstract category parameter, and contravariant in the
Physical category parameter. Dang, I hope some part of this makes any sense
:)



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