"Eilenberg-MacLane property" for the classifying space of a groupoid

Question

Given a groupoid $G$, its classifying space is defined as the standard geometric realisation of the nerve.

My question is: since the classifying space of a group is the only space up to homotopy that has certain homotopy groups, can something like that be said about the case of a groupoid? For example, do the homotopy groups of $BG$ have some relationship (obviously much less trivial) with the groupoid $G$?

And if not (as I suspect) is there some other "geometrical" invariant of the space $BG$ that yields back $G$ in some form (like the first homotopy group in the case when $G$ is a group)?

Thank you in advance.

Answer 1

For example, do the homotopy groups of $BG$ have some relationship (obviously much less trivial) with the groupoid $G$?

Actually, it is not much less trivial at all! A groupoid is uniquely determined, up to equivalence, by the groups corresponding to its connected components. Indeed, if $G$ is a groupoid and $G_0\subseteq G$ is a skeleton of $G$, then $G_0$ is a disjoint union of groups (considered as 1-object categories). Since the inclusion functor of a skeleton of a category is an equivalence of categories, this determines $G$ up to equivalence.

Now, the classifying space functor turns equivalences of categories into homotopy equivalences and disjoint unions of categories into disjoint unions of spaces. So, if $G$ is a groupoid, then $BG$ is homotopy equivalent to a disjoint union of classifying spaces of groups, namely the groups corresponding to a skeleton of $G$.

Or, to state things more canonically (without choosing a skeleton), the groupoid $G$ is naturally equivalent to the fundamental groupoid of $BG$, and $BG$ has no higher homotopy groups with respect to any basepoint. Explicitly, each object of $G$ determines a point of $BG$ and each morphism of $G$ determines a path in $BG$, and this defines a functor $G\to \pi_{\leq 1}(BG)$ which is an equivalence of categories.

Answer 2

It is helpful to consider the fundamental groupoid $\pi_1(X,S)$ of a space $X$ for a set of base points $S$. Similarly we consider a groupoid $G$ as given with its object set say $S$. Its classifying space $BG$ should thus be considered as having a set of base points $S$. Then $\pi_1(BG,S)\cong G$.

For more on this idea see the paper Modelling and computing homotopy types: I. Note that the homotopy groups $\pi_i(BG,x)$ at any base point $x$ are zero for $i >1$.