3.7 Curve objects and the dynamics of theory revision

The preceding geometry identifies two infinitesimal repair directions, but it does not yet say how either direction evolves. Cockett, Cruttwell, and Lemay develop differential equations directly in a tangent category by introducing a curve object: a structural analogue of the time line that can parameterize solutions and flows [ Cockett et al. , 2021 ] . This supplies a disciplined integration semantics for DIAL.

Let \((\mathcal M,T)\) be a tangent category with the products and tangent limits required for this construction. A vector field on an object \(M\) is a section

A parameterized dynamical system consists schematically of an object \(M\), a vector field \(V\), and an initial-state map \(g:X\to M\). Given a curve object \((C,c_0,c_1)\), a solution is a map

that assumes the initial value \(g\) at \(c_0\) and whose tangent in the \(C\)-direction agrees with \(V\). In ordinary smooth geometry one may take \(C=\mathbb R\), with \(c_0=0\) and \(c_1\) the constant unit vector field.

The categorical definition contains an important qualification. A curve object is preinitial among dynamical systems in every parameter context: solutions are unique when they exist, but existence is not automatic. A vector field is complete when solutions exist for every initial condition. Under the hypotheses of the theory, complete vector fields correspond to flows

Thus a curve object is not a trajectory and DIAL is not itself a curve object. The curve object supplies abstract time; \(M\) is the maintained theory-state object; and \(\gamma \) is a possible history of theory repair.

The involution-algebroid variant.

The definition above is the later, general formulation of curve-object dynamics. Burke and MacAdam use a related but differently packaged notion in their development of involution algebroids [ Burke and MacAdam , 2019 , Section 2.5 ] . Their complete curve object is a dynamical system

with a second distinguished point \(1_R:1\to R\), uniqueness of morphisms from it to any dynamical system, and a lifting condition for linear vector fields. More precisely, for a differential bundle locally trivial with respect to \(R\), a linear system over a base system with a complete solution must itself have a complete solution. The additional point is not used in their paper, but is retained in anticipation of defining the target map of a groupoid integrating an involution algebroid.

Cockett, Cruttwell, and Lemay subsequently separate these ingredients. Their curve object is preinitial in every parameter context, its distinguished vector field commutes with itself, and that field is complete. They then state linear completeness as an additional axiom: a linear vector field on a differential bundle is complete whenever its base vector field is complete. The two presentations should therefore not be conflated. The earlier variant foregrounds the linear lifting needed for algebroid transport; the later framework supplies the contextual theory of solutions, flows, and commuting dynamics used here.

For DIAL, both layers matter. The ordinary curve-object axioms parameterize the two local theory-state flows. Transport of elements in the side or core bundles additionally requires the appropriate linear-completeness doctrine. Indeed, Burke and MacAdam use their complete curve object to extend infinitesimal transport along variations of admissible algebroid paths to full transport. A future integration of DIAL should consequently declare which differential bundles are covered, whether completeness is global or partial, and whether the curve object also carries the unit and function-space structure required for path and homotopy objects.

. In this book, curve object refers by default to the contextual notion used for differential equations and flows. When algebroid transport is required, we additionally assume and name the relevant linear-completeness doctrine. This convention prevents uniqueness of trajectories from being mistaken for existence of the linear transports needed to integrate DIAL.