0.10 What the categorical theory must provide

The examples leave five mathematical obligations for the rest of the book.

1. A presentation language must state candidate entities, transformations, composites, and equations without requiring the whole world in advance.

2. An identification criterion must say when successive presentations count as learning the same world, especially when only a restricted family of questions can be asked.

3. An online semantics must represent the fact that information is revealed through a growing, action-dependent history.

4. A theory of coherent repair must distinguish a bad estimate inside a sound declaration from evidence that forces the declaration itself to change.

5. A persistence principle must explain why a learned construction should survive and transport to future problems.

Categories, functors, natural transformations, Kan extensions, nerves, horns, and tangent structure will be introduced because they answer these obligations, not as decoration imposed on the examples. Chapter 1 now develops that language. The chapters after it return to the infant’s problem in an abstract form: what can be identified from interaction, what is visible to a probe, how incomplete compositions are filled, and when a defect requires accommodation rather than further assimilation.