Editing Openai/696eca7b-acdc-800a-9ad9-5ebc9f477576 (section)

==== 3) Muscle memory = path dependence → a 2-category (or a category with annotations) ====

The key issue: two paths can land on the same (n,e,c)(n,e,c)(n,e,c) but feel different later (faster retraining, etc.). In ordinary categories, equal targets don’t remember paths.

===== Reasonable upgrade A: enrich morphisms with “history” =====

Let morphisms be pairs (w,σ)(w, \sigma)(w,σ) where:
* www is the word in generators h,s,ph,s,ph,s,p
* σ\sigmaσ is a trace (e.g., multiset of “marks retained”)

Composition combines traces (e.g., union, monoid addition). That’s a category enriched over a monoid (simple and very practical).

===== Reasonable upgrade B (more categorical): a 2-category =====
* Objects: fiber states (or fibers)
* 1-morphisms: training programs (words in h,s,ph,s,ph,s,p)
* 2-morphisms: rewrites between programs (equivalences / comparisons of histories)

Example: if under some conditions ppp then hhh behaves like hhh then ppp, you’d have a 2-cell

α:(p∘h)⇒(h∘p)\alpha: (p\circ h) \Rightarrow (h\circ p)α:(p∘h)⇒(h∘p)
but you might not always have it—path dependence is exactly “not all such α\alphaα exist”.

This is a clean way to represent “same endpoint, different internal preparedness.”