2 questions organized, typos corrected

I am studying orbifold fundamental group (or more generally orbifold homotopy groups). In a nutshell, my questions is: what are they intuitively? The following are more precise In what follows I give definitions and more precise questions. My definition of orbifold fundamental group is via classifying space of groupoid, which is explained in the next paragraph (so you may want to skip it if you know the definition).

Let $\mathcal{G}$ be a topological groupoid consisting a topological spaces $G_{0}$ of $objects$ and $G_{1}$ of $arrows$ together with usual continuous structure maps. Let $|\mathcal{G}|$ denote the associated topological space $G_{0}/G_{1}$. Let $G_{n}$ be the iterated fibered product $G_{n}=G_{1}\times_{s,t} G_{n-1}$. These $G_{n}$ have the structure of a simplicial manifoldtopological space, called the $nerve$ of $\mathcal{G}$. Face operads $d_{i}:G_{n}\rightarrow G_{n-1}$ for $i=0,\dots,n$ are given by $$d_{i}(g_{1},\dots,g_{n})=(g_{1},\dots,g_{i}g_{i+1},\dots,g_{n})$$ for $i=1,\dots,n-1$ and $$d_{0}(g_{2},\dots,g_{n})=(g_{2},\dots,g_{n}), \ \ d_{n}(g_{1},\dots,g_{n})=(g_{2},\dots,g_{n-1}).$$ The classifying space $B\mathcal{G}$ of $\mathcal{G}$ is then defined as $$B\mathcal{G}=\bigsqcup_{n}(G_{n}\times \Delta^{n})((d_{i}(g),x)\sim(g,\delta_{i}(x))),$$ where $\Delta^{n}$ is the topological $n$-simplex and $\delta_{i}:\Delta^{n-1}\rightarrow \Delta^{n}$ is the standard facemap.\

The $n$-th orbifold homotopy group of $\mathcal{G}$ based at $x\in |\mathcal{G}|$ is defined to be $$\pi_{n}^{orb}(\mathcal{G},x)=\pi_{n}(B\mathcal{G},y),$$ where $y\in G_{0}$ maps to $x$ under the quotient map $G_{0}\rightarrow |\mathcal{G}|$. My troubles

The following are firstly that I don't understand why this my questions:

1. Why is this a reasonable definition and secondly that I cannot compute $\pi_{n}^{orb}$ even for a simple orbifold based on the definition. ? Any manifold $M$ can be thought of topological groupoid via its chart i.e. $G_{0}=\bigsqcup_{i}U_{i}$ and $G_{1}=\bigsqcup_{i,j}U_{i}\times_{M} U_{j}$. It is not clear to me that the definition above reproduce $\pi_{n}(M)$.

For example

2. I am aware of an explicit description the orbifold fundamental groups of the orbifold Riemann surface $\Sigma_{g,n,k}$ of genus $g$ and $n$ orbifold points $p_{i}$ of order $k_{i}$: $$\pi_{n}^{orb}(\Sigma_{g,n,k}) =\langle \alpha_{i},\beta_{i},\sigma_{j} \ (1\le i \le g,1\le j \le n)\ | \ \sigma_{1}\dots\sigma_{n}\prod_{i=1}^{g}[\alpha_{i},\beta_{i}]=1,,\sigma_{i}^{k_{i}}=1\rangle$$ I have trouble connecting the definition above and Is it easy to see this explicit presentation . by the definition above?

3. It seems there are several ways to define the fundamental group of an orbifold, such as covering space etc. How should one understand orbifold fundamental groups?

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# How should one understand orbifold fundamental groups?

I am studying orbifold fundamental group (or more generally orbifold homotopy groups). In a nutshell, my questions is: what are they intuitively? The following are more precise definitions and questions.

Let $\mathcal{G}$ be a topological groupoid consisting a topological spaces $G_{0}$ of $objects$ and $G_{1}$ of $arrows$ together with usual continuous structure maps. Let $|\mathcal{G}|$ denote the associated topological space $G_{0}/G_{1}$. Let $G_{n}$ be the iterated fibered product $G_{n}=G_{1}\times_{s,t} G_{n-1}$. These $G_{n}$ have the structure of a simplicial manifold, called the $nerve$ of $\mathcal{G}$. Face operads $d_{i}:G_{n}\rightarrow G_{n-1}$ for $i=0,\dots,n$ are given by $$d_{i}(g_{1},\dots,g_{n})=(g_{1},\dots,g_{i}g_{i+1},\dots,g_{n})$$ for $i=1,\dots,n-1$ and $$d_{0}(g_{2},\dots,g_{n})=(g_{2},\dots,g_{n}), \ \ d_{n}(g_{1},\dots,g_{n})=(g_{2},\dots,g_{n-1}).$$ The classifying space $B\mathcal{G}$ of $\mathcal{G}$ is then defined as $$B\mathcal{G}=\bigsqcup_{n}(G_{n}\times \Delta^{n})((d_{i}(g),x)\sim(g,\delta_{i}(x))),$$ where $\Delta^{n}$ is the topological $n$-simplex and $\delta_{i}:\Delta^{n-1}\rightarrow \Delta^{n}$ is the standard facemap.\

The $n$-th orbifold homotopy group of $\mathcal{G}$ based at $x\in |\mathcal{G}|$ is defined to be $$\pi_{n}^{orb}(\mathcal{G},x)=\pi_{n}(B\mathcal{G},y),$$ where $y\in G_{0}$ maps to $x$ under the quotient map $G_{0}\rightarrow |\mathcal{G}|$. My troubles are firstly that I don't understand why this is a reasonable definition and secondly that I cannot compute $\pi_{n}^{orb}$ even for a simple orbifold based on the definition. Any manifold $M$ can be thought of topological groupoid via its chart i.e. $G_{0}=\bigsqcup_{i}U_{i}$ and $G_{1}=\bigsqcup_{i,j}U_{i}\times_{M} U_{j}$. It is not clear to me that the definition above reproduce $\pi_{n}(M)$.

For example I am aware of an explicit description the orbifold fundamental groups of the orbifold Riemann surface $\Sigma_{g,n,k}$ of genus $g$ and $n$ orbifold points $p_{i}$ of order $k_{i}$: $$\pi_{n}^{orb}(\Sigma_{g,n,k}) =\langle \alpha_{i},\beta_{i},\sigma_{j} \ (1\le i \le g,1\le j \le n)\ | \ \sigma_{1}\dots\sigma_{n}\prod_{i=1}^{g}[\alpha_{i},\beta_{i}]=1,,\sigma_{i}^{k_{i}}=1\rangle$$ I have trouble connecting the definition above and this explicit presentation.

It seems there are several ways to define the fundamental group of an orbifold, such as covering space etc. How should one understand orbifold fundamental groups? Thank you for your assistance.