lin-0062

4.1 Tangent structure

A tangent category, formally defined in Section 0.11, abstracts the behavior of tangent bundles without requiring every object to be a smooth manifold [ Cockett and Cruttwell , 2014b , Rosický , 1984 ] . Its tangent endofunctor is

\[ T:\mathcal C\longrightarrow \mathcal C, \]

with structure maps \(p,0,+,\ell ,c\) satisfying the fiber-power, additive, coherence, and vertical-universality axioms reviewed there.

For a smooth manifold \(M\), \(TM\) contains pairs \((x,v)\) of a point and a tangent direction. For a smooth map \(f:M\to N\),

\[ Tf(x,v)=(f(x),Df(x)[v]). \]

The categorical axioms preserve the compositional fact

\[ T(g\circ f)=Tg\circ Tf. \]

This is the elementary mechanism behind tangent factorization.

Definition 4.1 Tangent model

For a learning-sketch model \(D:J\to \mathcal C\), its tangent model is

\[ TD:J\longrightarrow \mathcal C, \qquad (TD)(j)=T(Dj),\quad (TD)(u)=T(Du). \]

The graphical calculus makes the formula \(TD=T\circ D\) literal: the \(D\)-wire is pasted to the \(T\)-wire. If \(e:D\Rightarrow D'\) is a coherent model repair, horizontal composition with the identity on \(T\) yields the tangent repair \(Te:TD\Rightarrow TD'\). Thus differentiation transports a typed repair; it does not erase the naturality obligations that made the repair coherent.

Tangent structure in the 2-categorical graphical language. The lift introduces another tangent level; the flip exchanges two tangent directions. The glyphs follow the calculus of Cockett and Cruttwell [ 2015 ]
Figure 4.1 Tangent structure in the 2-categorical graphical language. The lift introduces another tangent level; the flip exchanges two tangent directions. The glyphs follow the calculus of Cockett and Cruttwell [ 2015 ] .