Tangents, probes, and infinitesimal structure

tangent category with chosen tangent structure \(\mathbb T=(T,p,0,+,\ell ,c)\)

Encodes admitted infinitesimal variation and its coherent transport.

tangent functor and tangent object

Maps an object to its object of first-order directions.

\(n\)-fold fiber power \(TX\times _X\cdots \times _XTX\)

Collects tangent vectors sharing one base point; \(T_2X\) is the domain of addition.

\(n\)-fold iterate of \(T\)

Represents iterated tangent levels; it is not the fiber power \(T_nX\).

projection, zero, fiberwise addition, vertical lift, and canonical flip

Structural maps constrained by the tangent-category axioms.

tangent lift of \(f\)

Transports infinitesimal variation through a declared map.

infinitesimally perturbed realization

A probe-indexed family with base point \(D_0\).

first-order nilpotent relation

Synthetic differential-geometric encoding of first-order variation.

Weil algebra or probe object

Specifies the admitted order and shape of infinitesimal variation.

tangent direction or probe

A local variation whose semantics must be declared.

Lie bracket of vector fields or infinitesimal actions

Detects order-sensitive interaction and failure of involutive closure.

category of \(\mathcal C\)-valued database instances

Carries pointwise tangent structure when \(\mathcal C\) is a tangent category.

pointwise tangent lift of a database instance

Differentiates instance values while leaving the schema fixed.

syntactic tangent expansion of a database schema

Declares tangent objects, fields, brackets, and their constraints explicitly.

bracket factorization obstruction

Records failure of the visible database distribution to close.

anchor of a Lie algebroid

Maps abstract skills or generators to vector fields on a base manifold.

null or gauge subspace

Directions quotiented before diagnosis when they are irrelevant to the declared decision.

declared tangent metric as a bundle morphism

Converts tangent vectors to covectors; its inverse \(g^\sharp \), when admitted, selects intrinsic repair directions.

metric on obstruction fibers

Measures the residual of an inexact typed repair without defining the obstruction itself.

Natural LINCS tangent repair

Least intrinsic model variation that cancels, or approximately contracts, the linearized typed obstruction.

Fisher information metric

Information-geometric choice of \(g_\theta \) for a declared stochastic model and sampling law.