3.1 The ambient differential universe

Let \(\mathcal M\) be a category whose objects are structured theory states, models, or realizations and whose arrows are admissible structure-preserving transformations. A tangent structure equips \(\mathcal M\) with an endofunctor

and natural transformations expressing projection, zero, fiberwise addition, vertical lift, and interchange [ Cockett and Cruttwell , 2014 ] . For an object \(X\), the projection

organizes first-order variations over their base state.

It is tempting to call \(\mathcal M\) “the search space.” That phrase is useful only with care. A tangent category is a categorical universe containing many spaces and transformations, not one set of candidate answers. The tangent structure says which local variations are meaningful and how they compose. The application supplies the semantics: a tangent direction may vary an axiom weight, an intervention field, a simulator mechanism, an observer, or a local model.

The interpretation clause prevents a purely formal tangent vector from being mistaken for a meaningful conceptual change. It plays the same role as the observability caveat in Learning in Infinitesimal Non-Compositional Sketches (LINCS): structure must be connected to an observer and an experimental semantics before it can guide learning.