Let me start by saying that these ideas are not due to me. I overheard them in a seminar I attended recently (see Footnote).
There are many situations in which one is working with a compact Lie group $G$ and it becomes very convenient to make the following assumption: $G$ is connected with torsion-free fundamental group. I don't want to go into the details of these "situations" (because that's sort of irrelevant to my question) beyond mentioning that two equivalent conditions are i) that the centralizer of any element of $G$ is connected and ii) $G$ is connected and all central extensions by $U(1)$ split (see Lemma 4.1 "Loop Groups and Twisted $K$-theory I" by Freed, Hopkins, and Teleman.
Anyway, let $T$ be a maximal torus of $G$ and let $BT$ and $BG$ be the associated topological groupoids with a single object.
My question: is the condition "the compact group $G$ is connected with torsion-free fundamental group" equivalent to the condition "the morphism $$BT\rightarrow BG$$ is an effective epimorphism"?
Motivation for asking: Effective epimorphisms are a very fashionable thing to talk about these days, and it would be nice to recast some familiar old math in terms of them.
Disclaimer: There are many different ways to think about the objects $BT$, and $BG$. Perhaps the question is more sensible, for instance, if I consider the associated stacks on some appropriate category of topological spaces. So I ask you to interpret the symbols $BT$ and $BG$ (and the category in which they reside) as favorably as possible, so that the statement has the highest chance of being correct.
(Footnote: I was stumped by the ethics here. On the one hand I want to respect the privacy of the people in that seminar; on the other hand I want to cite them appropriately. I decided on the former since I can always add a citation later, and I can't "un-breach" their privacy. Any wisdom on this matter would be appreciated.)
