simplicial group is, naturally enough, a simplicial object in the category of groups. It is equivalently a simplicial set ${G_\bullet}$ such that each ${G_n}$ 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 ${\infty}$-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 ${\infty}$-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 ${\infty}$-groupoid with one object should be the same as an ${\infty}$-group. This is in fact true with the above notation. In other words, if we say that an ${\infty}$-groupoid is a Kan complex (as usual), and decide that an ${\infty}$-group is a simplicial group, then the ${\infty}$-groups are the “same” as the ${\infty}$-groupoids with one object.

Here the “same” means that the associated ${\infty}$-categories are equivalent. One way of expressing this is to say that there are natural model structures on ${\infty}$-groupoids with one object and simplicial groups, and that these are Quillen equivalent. This is a frequent way of expressing the idea that two ${\infty}$-categories (or at least, ${(\infty, 1)}$-categories) are equivalent. For instance, this is the way the ${\infty}$-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 ${\mathbf{SGrp}}$ of simplicial groups and on the category ${\mathbf{SSet}_0}$ 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 ${G}$ is a topological group, then there is a classifying space ${BG}$ and a principal ${G}$-bundle ${EG \rightarrow BG}$ such that ${EG}$ is contractible. It follows from this that for any CW-complex ${X}$, the homotopy classes of maps ${X \rightarrow BG}$ are in bijection with the principal ${G}$-bundles on ${X}$ (in fact, ${EG \rightarrow BG}$ is a universal bundle).

We will need the appropriate notion of a principal bundle in the simplicial setting. Let ${G_\bullet}$ be a simplicial group.

Definition 2 A ${G_\bullet}$simplicial set is a simplicial set ${X_\bullet}$ together with an action ${G_\bullet \times X_\bullet \rightarrow X_\bullet}$ satisfying the usual axioms. Thus, each ${X_n}$ is a ${G_n}$-set. There is a category ${\mathbf{SSet}_G}$ of ${G_\bullet}$-simplicial sets and ${G_\bullet}$-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 3 Let ${E_\bullet \rightarrow B_\bullet}$ be a map of ${G_\bullet}$-simplicial sets. We say that ${E_\bullet \rightarrow B_\bullet}$ is a principal ${G_\bullet}$-bundle if ${B_\bullet}$ has trivial action, each ${E_n}$ is a free ${G_n}$-set, and if ${E_\bullet/G_\bullet \simeq B_\bullet}$ under the natural map. (more…)