2.1.4 The tangent refinement

2.1.4 The tangent refinement

The unknown world may carry more than compositional structure. A tangent category equips \(\mathcal C\) with an endofunctor \(T\), projection and zero maps \(p:T\Rightarrow 1\) and \(0:1\Rightarrow T\), fiberwise addition, a vertical lift \(\ell :T\Rightarrow T^2\), and a canonical flip \(c:T^2\Rightarrow T^2\), subject to the tangent-category axioms reviewed in Chapter 1. We write such a world as

Its tangent objects do not merely append more observations. They declare which infinitesimal variations are meaningful, how they return to a base object, how they add, and how iterated variations cohere.

Let \(\mathbf{TanCat}_{\mathcal U}\) denote the 2-category of \(\mathcal U\)-small tangent categories, strong tangent functors, and tangent transformations. There is a forgetful 2-functor

Tangent ORACLE restricts its hypothesis space to a suitable sub-2-category \(\mathfrak H_{\mathrm{tan}}\) and seeks an unknown object \(\mathbb C^{\mathrm{tan}}_\star \) up to the equivalence exposed jointly by base and tangent probes.

This is a genuine strengthening of the learning problem. Identifying \(\mathcal C_\star \) does not in general identify a tangent structure over it: the fiber \(U_{\mathrm{tan}}^{-1}(\mathcal C_\star )\) may contain several inequivalent tangent doctrines. Conversely, a tangent prior can make learning easier by ruling out base-category hypotheses whose admissible variations do not fit the observed infinitesimal behavior.

Nor does identifying a tangent category necessarily identify a differential category behind it. One possible prior starts with a differential category \((\mathcal X,\otimes ,I,!,d)\) and exposes a tangent world through the coEilenberg–Moore category of \(!\)-coalgebras, provided the required limits exist [ Cockett et al. , 2020 ] . This is a differential realization of the tangent hypothesis, not extra structure that every tangent world must possess. Figure 1.7 therefore gives ORACLE two legitimate targets: the observable infinitesimal geometry, or a latent differential presentation that generates it.