The Whole Is a Function, the Parts Are Subfunctions——A Rigorous Proof in the Framework of Category Theory

Abstract

We present a rigorous proof, entirely within the framework of category theory, of the proposition that “the whole is a function, the parts are subfunctions”. Using the Yoneda embedding, any object $X$ (the “whole”) is identified with its representable functor $h_X = \Hom(X,-)$, which can be viewed as a “function”. For a subobject $i: A \hookrightarrow X$ (the “part”), the embedding induces a natural transformation $\res: h_X \to h_A$ whose components are restriction maps $f \mapsto f \circ i$. Thus the part corresponds to the functor $h_A$ and the restriction natural transformation precisely expresses that the part is a subfunction of the whole. This result is independent of any specific mathematical structure and reveals the fundamental functorial nature of the whole–part relation.

Keywords