ora-0110
8.7 Toward a categorical dimension theory
Finite cardinality is too crude for rich UOCL classes. The PAC analogue suggests a dimension, but arbitrary binary labelings of isolated points would again forget composition.
For a binary probe doctrine, a finite compatible family \(Q_0\subseteq \mathcal Q\) is categorically shattered by \(\mathbf H\) when every binary answer pattern that satisfies the declared typing, composition, and coherence constraints is realized by some hypothesis in \(\mathbf H\). The categorical probe dimension \(\mathsf{CPdim}(\mathbf H,\mathcal Q)\) is the supremum of the sizes of such families.
When \(\mathcal Q\) is discrete, compatibility imposes no additional relations and this reduces to the familiar shattering idea behind VC dimension. For general doctrines, deriving uniform-convergence bounds from \(\mathsf{CPdim}\), or replacing it by enriched covering numbers or a homotopical complexity invariant, is a theorem program rather than a result claimed here. The definition records the principle that complexity should count independently variable compositional obligations, not simply the cardinality of a presentation.