[FRIAM] A/R theory

[Frank Wimberly]

A partition, e.g. the buckets, defines an equivalence relation which is
transitive.  Or is that what you said.

---
Frank C. Wimberly
140 Calle Ojo Feliz,
Santa Fe, NM 87505

505 670-9918
Santa Fe, NM

On Sun, Nov 8, 2020, 7:25 PM jon zingale <jonzingale at gmail.com> wrote:

> 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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