Geometric Sub-bundles

 Let $\mathfrak{X}$ be a topological stack, and $LocSys(\mathfrak{X})$ a local system taking varieties $v\in \mathfrak{X}$ to their projective resolutions over an affine coordinate system. Let $\alpha$ and $\beta$ be smooth charts encompassing non-degenerate loci of the upper-half plane, and let $\phi$ be the map $\beta \circ {\alpha }^{-1}$. Our goal is to describe a class of vector bundles, called $\text{emph}geometricsub-bundles$, which provide holonomic transport for n-cells (for small values of n) over a $G_{\delta }$-space which models the passage $\mathfrak{X}\rightrightarrows LocSys(\mathfrak{X})$. We will first establish the preliminary definitions before advancing our core idea, which succinctly states that for a pointed, stratified space $Stra{t}_M^{\ast }$, there is a canonical selection of transition maps $[\phi ]$ which preserves the intersection of a countable number of fibers in some sub-bundle of the bundle $Bu{n}_V$ over $LocSys(\mathfrak{X})$.