ifc-0106

7.1 From juxtaposition to composition

Use the differential representational package introduced in Chapter 0:

\[ \mathfrak R_{\mathrm{diff}} =(\mathbb S,\mathfrak D,\mathcal M,F,\Lambda ,\mathcal G,\mathcal A), \]

where \(\mathbb S\) is a sketch or theory presentation, \(\mathfrak D\) its preservation doctrine, \(\mathcal M\) a semantic universe, \(F\) the selected Weil action, \(\Lambda \) the observer family, \(\mathcal G\) the generative operations, and \(\mathcal A\) the admission procedures. Suppose \(X\) and \(Y\) are already represented in this package. Placing descriptions of \(X\) and \(Y\) next to one another is a juxtaposition. A combination requires a registered way for information, constraints, or action to cross between them.

In the simplest case, the interface is already present as a pair of maps

\[ X \xleftarrow {i} I \xrightarrow {j} Y. \]

The object \(I\) names what the two domains share: a boundary state, a common variable, a unit convention, a protocol, or an abstract pattern. If the ambient doctrine admits the relevant colimit, a pushout may construct a glued object \(X\amalg _IY\). In another application the correct construction may be a pullback, tensor product, profunctorial composite, or ordinary substitution. The categorical form is not chosen for decoration; it states what is shared and which universal property the combination is expected to satisfy.

Definition 7.1 Typed creative combination

A typed creative combination of \(X\) and \(Y\), relative to \(\mathfrak R_{\mathrm{diff}}\), consists of an explicit interface, an admissible composition operation in the declared theory, and an observer-level claim about the composite. It becomes an admitted combination only when the composition obligations and the claimed consequence pass the registered tests \(\mathcal A\).

This definition separates four increasingly demanding cases:

  1. Juxtaposition: two items are mentioned together, with no map between them.

  2. Legal composition: a registered interface makes the construction well typed.

  3. Nontrivial interaction: the composite has an observable property not reducible to the two isolated descriptions.

  4. Productive combination: the interaction transfers, predicts, or reorganizes later constructions.

Only the last three are compositional achievements, and legality alone is a weak creativity criterion. A type checker can reject nonsense; it cannot certify significance.