lin-0109

8.5 Tangent lifting of parametrized modules

For \(f:P\times X\to Y\), ordinary tangent structure gives

\[ Tf:TP\times TX\longrightarrow TY. \]

After the canonical product identification, this is again a parametrized map, now with parameter object \(TP\). Hence tangent lifting descends to parametrized maps when it respects the chosen reparameterization equivalence.

Proposition 8.5 Tangent descent

If representatives \(f:P\times X\to Y\) and \(f':P'\times X\to Y\) differ by a diffeomorphic reparameterization \(\phi :P'\to P\), then their tangent representatives differ by \(T\phi :TP'\to TP\). Therefore \(T\) defines a well-typed lift on the corresponding parametrized arrows.

Proof ▼

Differentiate \(f'=f\circ (\phi \times \operatorname {id}_X)\) and use functoriality of \(T\).

Applying this lift to every arrow gives the tangent candidate

\[ T_{\mathbf{Para}}D: \mathsf{FMon}(\Sigma )\longrightarrow \mathbf{Para}. \]

Its factorization obstruction is the first Deep-LINCS signal:

\[ \mathcal O_1(D) = \mathcal O_{\mathbb S^\otimes }(T_{\mathbf{Para}}D). \]