TITLE: On fundamental groupoid of fundamental groupoid
QUESTION [7 upvotes]: Given a topological space $X$, we have the notion of the fundamental groupoid $\Pi_1(X)$. 
Here, the fundamental groupoid $\Pi_1(X)$ is made into a topological groupoid giving a topology on the morphism set.
One can then talk about $\Pi_1(\Pi_1(X))$. Is this related to (same as) the fundamental $2$-groupoid?

REPLY [4 votes]: The fundamental groupoid $\Pi_1(M)^{top}$, when equipped with the 'usual topology' when the space $M$ permits it, is weakly/Morita equivalent to the same groupoid equipped with the discrete topology, call it $\Pi_1(M)^\delta$. This is essentially Proposition 4.42 in my thesis. Even better, the identity functor $\Pi_1(M)^\delta \to \Pi_1(M)^{top}$ is a weak equivalence. 
If a functor $X\to Y$ of topological groupoids is a weak equivalence, then $\Pi_1(X) \to \Pi_1(Y)$ is an equivalence of groupoids. And lastly, for a topological groupoid $X$ that has the discrete topology, the canonical functor $X\to \Pi_1(X)$ is an equivalence. So for a suitable topological space $M$, there is a canonical functor $\Pi_1(X)^\delta \to \Pi_1(\Pi_1(M)^{top})$ and it is an equivalence.