# Standard way to prove that groupoids are homotopy 1-types

It is very well know that groupoids, considered as spaces via the nerve construction, are homotopy 1-types, i.e. aspherical. Here is a sketch of proof: Consider the canonical functor $f:C\rightarrow \pi_1(C)$ for a category $C$ and its fundamental groupoid $\pi_1(C)$. The Quillen-fiber based at some object $Y$, i.e. the slice category $Y\backslash f$, is the universal covering category of $C$. If $C=G$ is a groupoid, then $\pi_1(G)$ is $G$ itself and $f$ is essentially the identity. It is not hard to see that the Quillen-fiber in this case has a terminal object and thus is contractible, which proves the statement.

Is this the standard way to prove that groupoids have vanishing higher homotopy groups?

-
How about this? It is well-known that the nerve of a groupoid is a Kan complex, and that the nerve of any category is 2-coskeletal. But 2-coskeletal Kan complexes must have trivial higher homotopy groups. – Zhen Lin Sep 17 '13 at 9:18
What on earth do you mean by "the standard way "? – Tim Porter Sep 22 '13 at 7:18

## 1 Answer

A low tech-argument:

-a groupoid is a disjoint union of connected groupoids

-the nerve construction preserves disjoint unions

-a connected groupoid is equivalent to a group, and thus its nerve is aspherical.

-
Ah well, I know that a group being aspherical is even more standard than a groupoid being aspherical. But in the spirit of the question you should also give a nice argument for this case. ;) – Werner Thumann Sep 18 '13 at 9:05
A good construction of the classifying space BG of a group comes with a construction of a universal cover EG >--> BG with the discrete group G as fiber. Therefore BG = K(G,1). – Peter May Sep 18 '13 at 13:47