lin-0085

6.3 Vector fields as natural database sections

A tangent instance contains infinitesimal values but does not yet contain an infinitesimal intervention. A vector field on an instance \(I\) is a natural transformation

\[ V:I\Longrightarrow T_{\mathcal S}I \]

satisfying

\[ p_I\circ V=1_I. \]

For every schema arrow \(f:s\to t\), naturality requires

\[ T(I(f))\circ V_s = V_t\circ I(f). \]

Consequently, a database vector field is not an unrelated collection of per-table derivatives. Its components must respect the foreign keys and transformations declared by the schema.

Definition 6.4 Infinitesimal database

An infinitesimal database over \(\mathbb S\) is a tuple

\[ \mathfrak D_{\partial } = (\mathbb S,I,T_{\mathbb S}I,\mathcal V,\Lambda ), \]

where \(I\) is an admissible instance, \(T_{\mathbb S}I\) is its admitted tangent lift, \(\mathcal V\) is a registered family of natural vector fields, and \(\Lambda \) records the observers and provenance by which their values are estimated.

If tangent fibers have negatives and an estimated field fails naturality, the route difference

\[ \nu _f(V) = T(I(f))V_s - V_t I(f) \]

is a typed obstruction localized to the schema arrow \(f\). Without negatives, the ordered pair of routes and its failed equality play the same role; no subtraction is implied. This distinguishes two failures that an ordinary edge table would conflate: incompatibility of a field with data migration, and non-closure among fields that are already compatible.