0.9 Tangent categories and local possibility

Calculus describes local variation on numerical spaces. A tangent category abstracts the compositional structure required to speak of tangent objects and maps even when ordinary coordinates are not primary [ Cockett and Cruttwell , 2014 ] . At minimum, the reader should retain the following data.

The intended picture is simpler than the axioms. An object \(X\) represents a space of current possibilities; \(TX\) represents a possibility together with an admissible first-order way of varying it. A map \(f:X\to Y\) transports both the current possibility and its local variation through \(Tf:TX\to TY\). The additional structure below makes zero variation, addition, lifting, and the interchange of two successive variations compositional. Chapter 3 states the formal hypotheses; here the purpose is to make local experiments available outside ordinary coordinate spaces.

There is an endofunctor

sending \(X\) to a tangent object \(TX\), with a projection \(p_X:TX\to X\), a zero section, addition of tangent vectors over the same base point, a vertical lift, and a canonical flip on \(T^2X\). These maps obey coherence and universality axioms. The chain rule appears as functoriality:

The tangent category is not one search space. It is an ambient universe in which selected objects possess a declared notion of local variation. A tangent direction might vary a coefficient, a simulator mechanism, an observer, a skill, or a local causal field. The application must supply that meaning.

For the Cartesian circle

a first-order displacement \((\dot x,\dot y)\) tangent to the circle satisfies

The equation does not replace the circle. It probes which local variations preserve its defining relation. This pattern generalizes: declare a structure, differentiate its obligations, and use the resulting failure to localize where the current model is unstable.