Compact groupoid presentations for closed 2-orbifolds (or finite graphs of finite groups)?

Question

A result of Noohi says that if $\mathsf{X}$ and $\mathsf{Y}$ are topological stacks and $\mathsf{Y}$ admits a groupoid presentation $\mathcal{G}$ in which both $\mathcal{G}_0$ and $\mathcal{G}_1$ are compact, then the mapping stack $\operatorname{Map}(\mathsf{Y},\mathsf{X})$ is again a topological stack.

If $\mathsf{Y}$ is (the stack associated to) a closed, good 2-orbifold or a finite graph of finite groups, then $\mathsf{Y}$ is presented by the action groupoid of a finite group on a compact space (i.e. a closed surface or a finite graph), so Noohi's result applies.

My question arises from the observation that the "usual" ways to write down an étale groupoid representing either a closed 2-orbifold or a finite graph of finite groups do not yield groupoids where even the object space is compact. For example, for an orbifold, the object space is usually a disjoint union of open subsets of $\mathbb{R}^n$. If you're clever you can get away with finitely many open sets, but the open sets are certainly not compact.

My question is: is there a variation on the standard construction of a groupoid associated to an orbifold or a graph of groups that yields a compact space of objects and arrows in the case of a closed, good 2-orbifold or a finite graph of finite groups? I expect this groupoid is likely not étale; that's fine by me.

Even more concretely, consider the groupoid $\mathcal{G}$ whose object space is the disjoint union $[0,1] \sqcup [1,2]$ and which has exactly two nonidentity arrows identifying the copies of ${1}$. Is the stack $\mathsf{B}\mathcal{G}$ isomorphic to (the stack associated to) the closed interval?

Answer 1

For the "even more concretely" part, I think we can say that yes, $\mathsf{B}\mathcal{G}$ is isomorphic to the interval. Essentially because there is at most one arrow between any two points of $\mathcal{G}_0$, the orbit-space map $\mathcal{G}_0 \to [0,2]$ is a principal $\mathcal{G}$-bundle, defining by the Yoneda lemma a map $\underline{I} \to \mathsf{B}\mathcal{G}$. It seems pretty straightforward to show that this map is an isomorphism.

This suggests but does not prove that one could alter the usual definition of the étale groupoid $\mathcal{G}$ associated to an orbifold or a graph of groups by working with, for example, closed balls in $\mathbb{R}^n$ equipped with finite group actions, or closed balls in trees equipped with $\mathscr{G}_v$ actions, and then glue along overlaps as usual. The resulting groupoid $\bar{\mathcal{G}}$ is of course not étale, but will have (if you're careful) a compact space of objects and arrows in the case of a closed 2-orbifold or a finite graph of finite groups. If you take these balls to be contained in slightly larger open balls defining an étale groupoid presenting the orbifold or graph of groups, the inclusion $\iota\colon \bar{\mathcal{G}} \to \mathcal{G}$ is full, faithful and essentially surjective—at least, in a category-theoretic sense. Depending on your definition of equivalence, it may or may not be an equivalence of topological groupoids, essentially because that definition is catered towards étale (or merely open) groupoids, and $\mathcal{\bar{G}}$ is neither. Nevertheless, I believe it follows that $\mathsf{B}\bar{\mathcal{G}}$ and $\mathsf{B}\mathcal{G}$ are isomorphic as stacks, since the functor $\mathsf{B}$ is an equivalence of weak 2-categories? I'd love input on this last bit.