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.

Let $X$ be a reasonable space, so that the universal covering space $U(X)$ exists, and furthermore such that $U(X)\to X$ is a $\pi_1(X)$-principal bundle, where $\pi_1(X)$ is the fundamental group of $X$. The fundamental group $\pi_1(X)$ acts on $U(X)$ by deck transformations. Then, topologically, I believe that the fundamental groupoid $\Pi_1(X)$ (better notation than in the question, I think) is a $\pi_1(X)$-bundle $U(X) \hookrightarrow \Pi_1(X) \to X$ (using either the source or the target projection) associated to the $\pi_1(X)$ principal bundle $U(X)\to X$ with fiber $U(X)$.

The fundamental group $\pi_1(\Pi_1(X)) \cong \pi_1(X)$, as for any fibration with simply connected fibers (wiki). Iterating the above construction, you will see that $\Pi_1(\Pi_1(X)) \to \Pi_1(X)$ will be once again a $\pi_1(X)$-bundle, but with bigger fibers, since $U(\Pi_1(X)) \cong U(X) \times U(X)$. And so on. But, basically, the higher iterations $\Pi_1(\Pi_1(\cdots\Pi_1(X)))$ will not contain homotopical information about $X$ that's not already in $\pi_1(X)$ or $\Pi_1(X)$.

I mention that the question of topologising $\pi_1X$ when $X$ has a universal cover is generalised in 10.5.8 of the book Topology and Groupoids.

The question as a whole is relevant to the history of topology, see for example the book with that title edited by I.M. James (Elsevier, 1999).

In the early 20th century, topologists such as Dehn were looking for generalisations to all dimensions of the fundamental group $\pi_1(X,x)$ of a pointed space. At the 1932 ICM in Zurich, E. Cech gave a seminar on higher homotopy groups $\pi_n(X,x)$, and explained that they were abelian for $n \geqslant 2$. On this ground, Alexandroff and Hopf argued that they were not the hoped for generalisation, and only a small paragraph appeared in the ICM Proceedings. The importance of higher homotopy groups became clear later with work of Hurewicz. The historical side of this is discussed in this paper.

The argument for the abelian nature of higher homotopy groups is nowadays often expressed as "a group object in the category of groups is an abelian group".

The notion of groupoid was defined by Brandt in 1926, for applications in number theory, and many applications to differential geometry were developed by C. Ehresmann and his group. He also developed notions of $n$-fold groupoids as for instance for $n=2$ as groupoid objects in the category of groupoids. This led me in 1965 to look for "strict higher homotopy groupoids" and a higher Seifert-Van Kampen Theorem. This was found in dimension 2 in collaboration with P.J.Higgins in 1974, giving applications to second relative homotopy groups, relevant to J.H.C.Whitehead's 1949 paper "Combinatorial Homotopy II", and in all dimensions in 1981. That theory is developed not for pointed spaces but for filtered spaces. The argument for this is developed in this paper - essentially that to calculate something about a space you need information on it, and that information will have a certain structure which may be able to be formalised appropriately. The usual structure of a base point is rather minimalist for this end.