Given a topological space $X$, we have the notion of the fundamental groupoid $\Pi\_1(X)$. 


[Here](https://ncatlab.org/nlab/show/fundamental+groupoid), 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?



On fundamental groupoid of fundamental groupoid
The discussion at the linked nlab page points to a few different accounts of topologies on the fundamental groupoid, but I'm not feeling energetic enough to track them all down.


So let me just consider the simplest thing. Let $X$ be a topological space and consider the following topologically-enriched groupoid, which we'll call $\Pi\_1(X)$:


* The objects of $\Pi\_1(X)$ are the points of $X$.
* The morphisms of $\Pi\_1(X)$ are the usual morphisms of the fundamental groupoid, topologized as a quotient of a subspace of the exponential $X^I$ (in the compact-open topology).


Now consider the case where $X$ is a CW complex. In this case, $X^I$ is locally path-connected, and similarly so is the subspace of paths from $x$ to $y$ for any $x,y \in X$. So when we quotient by the appropriate equivalence relation -- i.e. take path components of this space of paths -- we get a discrete space. That is, $\Pi\_1(X)$ is just the discrete topologically-enriched groupoid on the ordinary fundamental groupoid.


In particular, $\Pi\_1(X)$ contains no information about higher homotopy. So even though I'm not sure what you mean by $\Pi\_1(\Pi\_1(X))$, there's no way you'll recover information about $\pi\_2(X)$.


It's in principle conceivable that for some class of wild spaces very different from CW complexes, there might be information about $\pi\_2(X)$ encoded in $\Pi\_1(X)$. But I think this is unlikely -- if there were a systematic way to understand $\pi\_2(X)$ in terms of $\Pi\_1(X)$, I would expect it to already be manifest in the tamest case where $X$ is a CW complex.


