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The mathematics of relation. Developed in the 1940s by Eilenberg and Mac Lane, category theory studies morphisms — arrows between things — rather than objects with internal properties. An object has no interior. It is known entirely by its arrows: how it relates to everything else.
A category consists of objects, arrows between them, a rule for composing arrows, and an identity arrow for each object.
The Yoneda Lemma is a central result. The totality of an object’s relationships to every other object in the category is a complete characterisation. What can be known about an object lives entirely in its relational profile.
P0 and P1 together establish the conditions that category theory formalises. Differentiation (P0) given relational character (P1) is categorical territory — morphisms, arrows, composition.
The Yoneda Lemma and P2 express the same conclusion through different languages. Yoneda: what can be known about an object is exhausted by its relationships. P2: the subject accesses reality only through the relational. One arrived at mathematically, the other philosophically. The convergence is structural.
A functor cannot create a morphism in the target that has no structural basis in the source. Higher categories add layers of expression about relations already present — they make visible what was implicit. They do not add relational power. This converges with P5: complexity grows in expression, not in power. The full power was always there.
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