0.3 What is a compositional world?

The word world does not mean the total physical universe. It means a stable domain of distinctions and transformations available to a learner. Natural language is one such world; embodied manipulation is another; social interaction is a third. They overlap, but none is reducible to a single input-output function.

For now, we use four informal questions to recognize a compositional world:

1. What are the recurring entities or states that the learner treats as stable enough to revisit?

2. Which transformations connect them, and when can two transformations be performed in sequence?

3. Which different histories count as equivalent for prediction or action?

4. Which missing constructions must be supplied to answer a new question, and which contradictions require the organization itself to change?

These questions foreshadow objects, morphisms, composition, equations, universal properties, and repair. They do not assume that there is one unique category hidden behind every domain. A categorical model is a declaration of which distinctions and composites matter to a specified family of probes. Two declarations may organize the same interaction at different scales.