A *simplicial group* is, naturally enough, a simplicial object in the category of groups. It is equivalently a simplicial set such that each has a group structure and all the face and degeneracy maps are group homomorphisms. By a well-known lemma, a simplicial group is automatically a Kan complex. We can thus think of a simplicial group in two ways. On the one hand, it is an -groupoid, as a Kan complex. On the other hand, we can think of a simplicial group as a model for a “higher group,” or an -group. This is the intuition that the nLab suggests.

Now it is well-known that in ordinary category theory, a group is the same as a groupoid with one object. So an -groupoid with one object should be the same as an -group. This is in fact true with the above notation. In other words, if we say that an -groupoid is a Kan complex (as usual), and decide that an -group is a simplicial group, then the -groups are the “same” as the -groupoids with one object.

Here the “same” means that the associated -categories are equivalent. One way of expressing this is to say that there are natural *model structures* on -groupoids with one object and simplicial groups, and that these are Quillen equivalent. This is a frequent way of expressing the idea that two -categories (or at least, -categories) are equivalent. For instance, this is the way the -categorical Grothendieck construction is stated in HTT, as a Quillen equivalence between model categories of “right fibrations” and simplicial presheaves. Let us express this formally.

Theorem 1 (Kan)There are natural model structures on the category of simplicial groups and on the category of reduced simplicial sets (i.e. those with one vertex), and the two are Quillen equivalent.

To give this construction, we will first describe the *classifying space* of a simplicial group. One incarnation of this is going to give the functor from simplicial groups to reduced simplicial sets. This is a simplicial version of the usual topological classifying space. Recall that if is a topological group, then there is a *classifying space* and a principal -bundle such that is contractible. It follows from this that for any CW-complex , the homotopy classes of maps are in bijection with the principal -bundles on (in fact, is a *universal* bundle).

We will need the appropriate notion of a principal bundle in the simplicial setting. Let be a simplicial group.

Definition 2A –simplicial setis a simplicial set together with an action satisfying the usual axioms. Thus, each is a -set. There is a category of -simplicial sets and -equivariant maps.

We next give the definition of a principal bundle. Notice that because of the combinatorial nature of simplicial sets, we don’t make any local trivialty condition; that will fall out.

Definition 3Let be a map of -simplicial sets. We say that is aprincipal -bundleif has trivial action, each is a free -set, and if under the natural map. (more…)