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

=== Now we upgrade from category to 2-category. ===

In a 2-category we have:
* Objects: fibers
* 1-morphisms: training transformations
* 2-morphisms: equivalences between histories of transformation

Example:

You can reach M₂ two ways:

Path A:

<syntaxhighlight>M₀ → M₁ → M₂

</syntaxhighlight>

Path B:

<syntaxhighlight>M₀ → M₀' → M₂

</syntaxhighlight>

Even if both end at same M₂, they may differ in:
* retained nuclei
* chromatin accessibility
* response speed to retraining

So we define a 2-morphism:

<syntaxhighlight>α : (g ∘ f) ⇒ (h ∘ k)

</syntaxhighlight>

Meaning:

: 

This is how muscle memory becomes path-dependent structure, not just state.

That is exactly the distinction between:
* A set
* A category
* A 2-category