ora-0034

2.1.1 From bare categories to structured worlds

Choose Grothendieck universes \(\mathcal U\in \mathcal V\). The hypothesis space cannot literally be the category of all categories: size must first be controlled. The 2-category \(\mathbf{Cat}_{\mathcal U}\) of \(\mathcal U\)-small categories is an object only at the larger universe level \(\mathcal V\).

Size discipline alone gives a prior that is far too weak. It says that the world has objects and composable arrows, but says nothing about which distinctions make fragmentary experience learnable. We therefore posit a small categorical sketch

\[ \mathbb T_{\mathrm{core}}, \]

whose sorts, arrows, specified diagrams, and designated cones express the learner’s initial organizing capacities. A realization in a candidate world \(\mathcal C\) is a structure-preserving interpretation

\[ M_{\mathcal C}:\mathbb T_{\mathrm{core}}\longrightarrow \mathcal C. \]

Here “structure-preserving” is relative to the declarations in the sketch; it does not mean that every imaginable limit or colimit must be preserved. When a fragment includes an observer into an auxiliary category, this notation abbreviates the corresponding multi-sorted diagram of categories rather than forcing every observer target to lie inside \(\mathcal C\).

Definition 2.1 Core-structured world

A core-structured world is a pair \((\mathcal C,M_{\mathcal C})\), where \(\mathcal C\) is a \(\mathcal U\)-small category and \(M_{\mathcal C}:\mathbb T_{\mathrm{core}}\to \mathcal C\) interprets the declared core-knowledge sketch. A morphism of core-structured worlds is a functor equipped with coherent comparison data preserving the declared core structure. Structured natural transformations are its 2-morphisms.

Write \(\mathbf{Cat}_{\mathcal U}^{\mathrm{core}}\) for the resulting 2-category and

\[ U:\mathbf{Cat}_{\mathcal U}^{\mathrm{core}} \longrightarrow \mathbf{Cat}_{\mathcal U} \]

for the forgetful 2-functor. The ORACLE hypothesis space is now a sub-2-category

\[ \mathfrak H_{\mathrm{core}} \subseteq \mathbf{Cat}_{\mathcal U}^{\mathrm{core}}, \]

and the true environment is an unknown structured object

\[ (\mathcal C_\star ,M_\star )\in \mathfrak H_{\mathrm{core}}. \]

The bare category \(\mathcal C_\star \) is only its image under \(U\). Thus the categorical prior is not a guess about the inventory of the world. It is a restriction on the kinds of organization through which an inventory can be discovered.

Guiding question.

Which part of \(\mathbb T_{\mathrm{core}}\) must be fixed for identification to be possible, which part is merely observational, and which part may itself be revised through accommodation?