This is same as saying a connected groupoid is equivalent to a group. But I have no idea how to construct such a group, can't even get started.
Any help is appreciated.
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$\begingroup$ What do you mean by equivalent? $\endgroup$ – Dori Bejleri Oct 15 '14 at 2:16
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1$\begingroup$ equivalence of categories $\endgroup$ – user112564 Oct 15 '14 at 2:23
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Let $\mathcal{Gr}$ be your groupoid category. Take any object $X \in \mathcal{Gr}$. Consider the subcategory $G$ consisting of just the single object $X$ and with $\operatorname{Hom}_G(X,X) = \operatorname{Hom}_\mathcal{Gr}(X,X)$. The category $G$ is a group because every element of $\operatorname{Hom}_G(X,X)$ can be composed by any other element, they're all invertible because $\mathcal{Gr}$ is a groupoid, and there is an identity morphism $X \to X$.
Now consider the inclusion functor $G \hookrightarrow \mathcal{Gr}$. This functor is fully faithful by construction since $\operatorname{Hom}_G(X,X) = \operatorname{Hom}_\mathcal{Gr}(X,X)$, and it is essentially surjective since $\mathcal{Gr}$ is is connected so for any $Y \in \mathcal{Gr}$, there exists a morphism $X \to Y$ which is necessarily an isomorphism since $\mathcal{Gr}$ is a groupoid. Therefore the inclusion is an equivalence of categories.
answered Oct 15 '14 at 2:36
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1$\begingroup$ Thanks. Did you use some theorems about equivalence of categories? Like essentially surjective + fully faithful implies equivalence? I don't know any of these theorems so I am a little puzzling.. $\endgroup$ – user112564 Oct 15 '14 at 5:02
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4$\begingroup$ @user112564 Yes, (assuming the axiom a choice) a functor is an equivalence iff it is fully faithful and essentially surjective. See this for example: en.wikipedia.org/wiki/… $\endgroup$ – Najib Idrissi Oct 15 '14 at 7:30
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Using a different terminology, but basically similar reasoning as Dori:
Every category is equivalent to its skeleton
Every connected groupoid is a category
In a skeleton you fuse together all isomorphic objects, therefore the skeleton of a connected groupoid is a category with a single element and invertible morphisms (a group)
So every connected groupoid is equivalent to a group
