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Robbie Lyman
Robbie Lyman
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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 not 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.

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 not may 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.

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.

Source Link
Robbie Lyman
Robbie Lyman
  • 2k
  • 10
  • 22

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 not may 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.