${\mathfrak{U}} = \{U_{\alpha} \}_{\alpha \in I}$ of $X$ and trivializations $$h_{\alpha}: U_{\alpha}^{V} \cong {(U_{\alpha})_{{+}}}$$ of $V$ over ${\mathfrak{U}}$ give equivalences $${A_{X}} (U_{\alpha}) = A^{{( U_{\alpha})_{{+}}}}
\xrightarrow{A^{h_{\alpha}}} A^{(U_{\alpha}^{V})} =
{A_{V}} (U_{\alpha}).$$ Over an intersection $U_{\alpha\beta} = U_{\alpha}\cap U_{\beta}$, we have an equivalence of ${A_{X}} (U_{\alpha\beta})$-modules $$g_{\alpha\beta} : {A_{X}} (U_{\alpha\beta}) {\xrightarrow{h_{\alpha}}}
{A_{V}} (U_{\alpha\beta}) {\xrightarrow{h_{\beta}^{-1}}}
{A_{X}} (U_{\alpha\beta}).$$

This analysis suggests that we let ${{\underline{A}}^{\times}}$ be the presheaf on spaces which on $Y$ consists of ${{\underline{A}}} (Y)$-module equivalences $${{\underline{A}}} (Y) \rightarrow {{\underline{A}}} (Y)$$ Then the $g_{\alpha\beta}$ are a Cech $1$-cocycle for the cover ${\mathfrak{U}}$, with values in the presheaf ${{\underline{A}}^{\times}}$, whose class $$[V] \in H^{1} (X;{{\underline{A}}^{\times}})$$ depends only on (the isomorphism class of) the vector bundle $V$. However $[V]$ is defined, it should be both the Cech cohomology class classifying the invertible ${A_{X}}$-module ${A_{V}}$ and the obstruction to giving a Thom isomorphism for $V$ in $A$-theory.

The difficulty in making this program precise is at once subtle and familiar, and arises from the suppleness of the presheaf ${\underline{A}}.$ For example, in trivializing the cocycle $g_{\alpha\beta},$ one expects to encounter a family of sections $$f^{0}_{\alpha} \in {{\underline{A}}^{\times}} ( U_{\alpha})$$ *and homotopies* $$\label{eq:37}
    f^{1}_{\alpha\beta}: f^{0}_{\beta}{\simeq}f^{0}_{\alpha}
g_{\alpha\beta}$$ in ${{\underline{A}}^{\times}} (U_{\alpha\beta}\times I).$ The $f^{1}_{\alpha\beta}$ will satisfy a coherence 