A steady map $f:Xto Y$ and a vector bundle $Eto X$ appear to present rise to a presheaf of groupoids on $Y$ alongside the next strains. For an open $Usubseteq Y$, every part of $f$ over $U$ (i. e. a map $s:Uto X$ with $fs=$ inclusion) produces a vector bundle $s^*E$ on $U$ and for one more part $t:Uto X$, a homotopy class $H$ of homotopies between $s$ and $t$ that induce trivial homotopy between $fs$ and $ft$ offers rise to a bundle isomorphism between $s^*E$ and $t^*E$.
Query 0: can this finalized to certainly yield a presheaf of groupoids?
Query 1 (in case the reply to the earlier one is “sure”): is that this a stack? Are there some pure restrictions on $f$ and $E$ (like $f$ being some type of fibration, and so forth.) which be sure that one will get a stack? Or perhaps some modifications of the development? For instance, as an alternative of demanding that $H$ induces a trivial homotopy, one may make a part of the construction a selection of a homotopy between $fH:fssim ft$ and the trivial homotopy $fs = $ inclusion $ = ft$.
Query 2: is that this an occasion of some identified building?