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.

 

A simple example of such a bundle is given by the trivial bundle {E_\bullet = B_\bullet \times G_\bullet}. Note that if {G} is a topological group and {E \rightarrow B} a principal {G}-bundle of topological spaces, then {\mathrm{Sing} E_\bullet \rightarrow \mathrm{Sing} B_\bullet} is a principal bundle over {\mathrm{Sing} G_\bullet}. The singular simplicial set functor sends principal bundles to principal bundles. The same also holds for the geometric realization functor, but this is harder.

Let {E_\bullet \rightarrow B_\bullet} be a principal bundle. We now want to make some kind of local trivialty assertion. Note that the pull-back of a principal bundle under any map {B'_\bullet \rightarrow B_\bullet} automatically becomes a principal bundle over {B'_\bullet}, so one way of expressing local trivialty is that there is a surjection {B'_\bullet \twoheadrightarrow B_\bullet} such that the principal bundle {E_\bullet \rightarrow B_\bullet} becomes trivial when pulled back to {B'_\bullet}. The claim, in fact, is that we taken just take {B'_\bullet} to be a disjoint union of simplices.

 

Proposition 4 (Local trivialty) A principal {G_\bullet}-bundle over a standard simplex {\Delta[n]_\bullet} is the trivial bundle.

Proof: Let {E_\bullet \rightarrow \Delta[n]_\bullet} be a principal {G_\bullet}-bundle. We can find a section {\Delta[n]_\bullet \rightarrow E_\bullet} by choosing any {n}-simplex of {E_\bullet} lying over the nondegenerate one of {\Delta[n]_\bullet}. But it is clear that any principal bundle with a section is trivial, by essentially the same proof as in the topological case. \Box

 

Corollary 5 Any principal {G}-bundle is a Kan fibration.

Proof: Indeed, we need only verify this when the base is a {\Delta[n]_\bullet} (note: a map {p: E_\bullet \rightarrow B_\bullet} is a Kan fibration if and only if every base-change via a simplex {\Delta[n]_\bullet \rightarrow B} is a Kan fibration). In this case, the principal {G}-bundle becomes {G_\bullet \times \Delta[n]_\bullet \rightarrow \Delta[n]_\bullet}. Since {G_\bullet} is a Kan complex, the result is clear. \Box

Now we want to show that there is a universal such bundle. To get one, it will be convenient to introduce a model structure on the category {\mathbf{SSet}_G} of {G_\bullet}-sets.

Definition 6 A map {X_\bullet \rightarrow Y_\bullet} of {G_\bullet} sets is a fibration (resp. weak equivalence) if and only if the associated map of ordinary simplicial sets is one. A cofibration in {\mathbf{SSet}_G} is a map with the left lifting property with respect to all fibrations which are weak equivalences.

 

It is not too difficult to check that this is a model structure on {\mathbf{SSet}_G}. Indeed, the point is that one can transfer a model structure along an adjunction. If {\mathcal{C}, \mathcal{D}} are compplete and cocomplete categories, and {F, G: \mathcal{C} \leftrightarrows \mathcal{D}} is an adjunction, then under suitable hypotheses a model structure on {\mathcal{C}} can be lifted to one on {\mathcal{D}}; one stipulates that a map {f} in {\mathcal{D}} is a fibration or weak equivalence if and only if {Gf } is. One such statement is in the Goerss-Schemmerhorn notes on model categories.

 

Theorem 7 Let {F, G: \mathcal{C} \leftrightarrows \mathcal{D}} be an adjunction between complete, cocomplete categories. Suppose {\mathcal{C}} is equipped with a cofibrantly generated model structure. Then the above definition of fibrations and weak equivalences in {\mathcal{D}} (by transfer along {\mathcal{D}}) endows {\mathcal{D}} with a model structure (where the cofibrations are forced) if:

  1. {G} commutes with sequential colimits.
  2. Any map in {\mathcal{D}} with the left lifting property with respect to all fibrations is a weak equivalence.

The pair {(F, G)} becomes a Quillen adjunction.

 

Let us apply this observation to the categories {\mathbf{SSet}, \mathbf{SSet}_G}. There is an adjunction

\displaystyle F, G: \mathbf{SSet} \leftrightarrows \mathbf{SSet}_G

where the left adjoint {F} sends a simplicial set {A_\bullet} to {A_\bullet \times G_\bullet}, and the right adjoint just forgets the {G_\bullet}-action. The above hypotheses of the theorem are satisfied, though we should show that any map in {\mathbf{SSet}_G} with the left lifting property with respect to the fibrations is a weak equivalence.

So, let {X_\bullet \rightarrow Y_\bullet} be a morphism in {\mathbf{SSet}_G} with the left lifting property with respect to all fibrations. We need to show that it is a weak equivalence. In other words, we need to show that it is a weak equivalence in {\mathbf{SSet}}. One can give a straightforward direct argument for this using the fact that the left adjoint preserves trivial cofibrations and {\mathbf{SSet}} is left proper, as follows. By the small object argument, we can factor {X_\bullet \rightarrow Y_\bullet} as a composite {X_\bullet \rightarrow Z_\bullet \rightarrow Y_\bullet} where the first map is a transfinite composite of push-outs of maps of the form {G_\bullet \times \Lambda_n^k \rightarrow G_\bullet \times \Delta[n]_\bullet}, and {Z_\bullet \rightarrow Y_\bullet} is a fibration. The first map {X_\bullet \rightarrow Z_\bullet} is clearly a cofibration in {\mathbf{SSet}_G} (by adjointness), and it is also easily seen to be a weak equivalence of simplicial sets because each {G_\bullet \times \Lambda_n^k \rightarrow G_\bullet \times \Delta[n]_\bullet} is (it is here we use left properness). By the retract argument, we find that {X_\bullet \rightarrow Y_\bullet} is a retract of {X_\bullet \rightarrow Z_\bullet} (because {X_\bullet \rightarrow Y_\bullet} has the left lifting property with respect to {Y_\bullet \rightarrow Z_\bullet}), and this establishes that the map is a weak equivalence.

In fact, even more is true:

Theorem 8 {\mathbf{SSet}_G} becomes a simplicial model category with the cofibrations, weak equivalences, and fibrations as above.

The generating cofibrations in {\mathbf{SSet}_G}, by construction more or less, are of the form {G_\bullet \times \partial \Delta[n]_\bullet \rightarrow G_\bullet \times \Delta[n]_\bullet}. One consequence of this:

 

Proposition 9 A cofibrant object in {\mathbf{SSet}_G} is precisely a {G_\bullet}-simplicial set {X_\bullet} such that {X_n} is a free {G_n}-set for each {n}.

The proof is relatively straightforward, and can be found in Goerss-Jardine.

 

Definition 10 classifying space {BG_\bullet} for {G_\bullet} is constructed as follows:

  1. Take a cofibrant {G_\bullet}-set {EG_\bullet} which is also a contractible Kan complex.
  2. Set {BG_\bullet = EG_\bullet/G_\bullet}.

 

Note that we have required that {EG_\bullet} be cofibrant, which is equivalent to saying that each level is a free {G_n}-set. It follows from this that {EG_\bullet \rightarrow BG_\bullet} is a principal {G_\bullet}-bundle. Also, we have required that {EG_\bullet} be a contractible Kan complex, as in the classical theory. By the general theory of model categories, we can do this: just take {EG_\bullet} as a cofibrant-fibrant approximation to {\ast}.

Note as one consequence of this, {BG_\bullet} is itself a Kan complex. This is a special case of the following easy fact: if {X_\bullet \rightarrow Y_\bullet} is a surjective morphism of simplicial sets, with {X_\bullet} a Kan complex, then {Y_\bullet} is one too.

We now claim that

Theorem 11 (Classification of principal bundles) Let {EG_\bullet \rightarrow BG_\bullet} be as above. Then for any simplicial set {X_\bullet}, pull-back of this bundle induces an isomorphism

\displaystyle [X_\bullet, BG_\bullet] \simeq G_\bullet-\mathrm{bundles \ on \ } X_\bullet

Proof: We shall start by proving the following weaker fact. If {E_\bullet \rightarrow Y_\bullet} is any principal {G_\bullet}-bundle and {f,g: X_\bullet \rightrightarrows Y_\bullet} are simplicially homotopic maps, then the pull-backs {f^*(E_\bullet), g^*(E_\bullet)} are isomorphic as bundles over {X_\bullet}. One easily reduces to the case {Y_\bullet = X_\bullet \times \Delta[1]_\bullet}. Consider the bundle {p: E_\bullet \rightarrow X_\bullet \times \Delta[1]_\bullet}. Let {E_{0 \bullet}} be the pull-back to {X_\bullet \times \left\{0\right\}}. We are going to show that {E_{0 \bullet } \times \Delta[1]_\bullet} and {E_{\bullet}} are isomorphic bundles over {X_\bullet \times \Delta[1]_\bullet}; this, as in the usual proofs in standard topology, will imply the result.

We have a diagram

The first horizontal map is the inclusion, and the second horizontal map is projection. In this diagram, the left map is a trivial cofibration: indeed, this follows from the freeness of the simplices in {E_{0 \bullet}}. The right map, as a bundle map, is a fibration. The lifting thus exists. Since the lifting is a morphism of bundles over {X_\bullet \times \Delta[1]_\bullet}, it is necessarily an isomorphism.

With this in mind, we can prove the theorem itself. We have just seen that there is indeed a natural transformation from {[X_\bullet, BG_\bullet]} into the set of isomorphism classes of bundles over {X_\bullet}. We need to show it is both injective and surjective. For the second, let {E_\bullet \rightarrow X_\bullet} be any bundle; we have to realize it as a pull-back of the bundle {EG_\bullet \rightarrow BG_\bullet}. To do this, we note that there is a map of {G_\bullet}-sets {E_\bullet \rightarrow EG_\bullet}; we find one such by lifting in the diagram

and using the model structure. Quotienting by {G_\bullet} gives a map {X_\bullet \rightarrow BG_\bullet} that fits into a commutative diagram

which must be cartesian, because we are dealing with principal bundles. This proves that the natural transformation above is surjective.

Next, we need to see that it is injective. But this follows because the map above {E_\bullet \rightarrow EG_\bullet} is unique up to equivariant homotopy (again by the model structure) and any equivariant homotopy leads to a homotopy on the quotient. \Box

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