of glued map space (as an ambient space of interest) jumps at $a=0$. We can find a linear map $\ominus_a$ such that $(\oplus_a,\ominus_a)$ is an isomorphism. So that we can parametrize geometric glued cylinder space $\bigsqcup_a\text{im}\oplus_a$ by $\bigsqcup_a\ker\ominus_a=r(U)$ realizeable as a retract, where maps in $r(U)$ have the same domain to compare. So $\bigsqcup_a\ker\ominus_a$ provides a local model as a retract for space $\bigsqcup_a\text{im}\oplus_a$ which is dimension varying. We can use a retract built from this plus other information as a parameter space of a good uniformizer of $B(Q,\omega)$ centered at a smooth map such that the universal property holds sc-smoothly. This says a good uniformizer is an **M-polyfold** (a paracompact and second countable space build with retracts as local models with sc-diffeomorphic coordinate changes, and here we just have a single chart).

We can also keep track of symmetry using étale proper Lie groupoid language based on M-polyfold with a generalization of properness ([@PolyfoldIII] 2.2) and such a groupoid is called an **ep-groupoid**. From properties of $B(Q,\omega)$ and sc-smooth universal property of good uniformizers, one can construct an ep-groupoid $B$ such that the orbit space $\underline{B}$ (modulo isomorphisms) is identified with $B(Q,\omega)$. A Morita-equivalence class of an ep-groupoid is 