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

==== 6) Local-in-time effects → a sheaf-y model (the sane version) ====

Training effects are local in time and then “glue” into a long-term adaptation narrative.

Let time be a topological space or just a poset of intervals Int\mathrm{Int}Int ordered by inclusion.

Define a presheaf FFF assigning to each interval UUU:
* F(U)F(U)F(U) = set (or category) of feasible training-response trajectories on UUU

Restriction maps:
* if V⊆UV\subseteq UV⊆U, then F(U)→F(V)F(U)\to F(V)F(U)→F(V) is “forget the outside portion”.

A sheaf condition says: if you have compatible local trajectories on overlapping subintervals, there is a unique glued global trajectory.

Muscle memory breaks uniqueness: you often get multiple global gluings consistent with local observations. That suggests “sheaf of groupoids” or “stack” (multiple consistent global histories).

That’s the useful upshot: memory = non-uniqueness of gluing.