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 2 A –simplicial set is 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 3 Let be a map of -simplicial sets. We say that is a principal -bundle if has trivial action, each is a free -set, and if under the natural map.
A simple example of such a bundle is given by the trivial bundle . Note that if is a topological group and a principal -bundle of topological spaces, then is a principal bundle over . 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 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 automatically becomes a principal bundle over , so one way of expressing local trivialty is that there is a surjection such that the principal bundle becomes trivial when pulled back to . The claim, in fact, is that we taken just take to be a disjoint union of simplices.
Proposition 4 (Local trivialty) A principal -bundle over a standard simplex is the trivial bundle.
Proof: Let be a principal -bundle. We can find a section by choosing any -simplex of lying over the nondegenerate one of . But it is clear that any principal bundle with a section is trivial, by essentially the same proof as in the topological case.
Corollary 5 Any principal -bundle is a Kan fibration.
Proof: Indeed, we need only verify this when the base is a (note: a map is a Kan fibration if and only if every base-change via a simplex is a Kan fibration). In this case, the principal -bundle becomes . Since is a Kan complex, the result is clear.
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 of -sets.
Definition 6 A map of sets is a fibration (resp. weak equivalence) if and only if the associated map of ordinary simplicial sets is one. A cofibration in 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 . Indeed, the point is that one can transfer a model structure along an adjunction. If are compplete and cocomplete categories, and is an adjunction, then under suitable hypotheses a model structure on can be lifted to one on ; one stipulates that a map in is a fibration or weak equivalence if and only if is. One such statement is in the Goerss-Schemmerhorn notes on model categories.
Theorem 7 Let be an adjunction between complete, cocomplete categories. Suppose is equipped with a cofibrantly generated model structure. Then the above definition of fibrations and weak equivalences in (by transfer along ) endows with a model structure (where the cofibrations are forced) if:
- commutes with sequential colimits.
- Any map in with the left lifting property with respect to all fibrations is a weak equivalence.
The pair becomes a Quillen adjunction.
Let us apply this observation to the categories . There is an adjunction
where the left adjoint sends a simplicial set to , and the right adjoint just forgets the -action. The above hypotheses of the theorem are satisfied, though we should show that any map in with the left lifting property with respect to the fibrations is a weak equivalence.
So, let be a morphism in 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 . One can give a straightforward direct argument for this using the fact that the left adjoint preserves trivial cofibrations and is left proper, as follows. By the small object argument, we can factor as a composite where the first map is a transfinite composite of push-outs of maps of the form , and is a fibration. The first map is clearly a cofibration in (by adjointness), and it is also easily seen to be a weak equivalence of simplicial sets because each is (it is here we use left properness). By the retract argument, we find that is a retract of (because has the left lifting property with respect to ), and this establishes that the map is a weak equivalence.
In fact, even more is true:
Theorem 8 becomes a simplicial model category with the cofibrations, weak equivalences, and fibrations as above.
The generating cofibrations in , by construction more or less, are of the form . One consequence of this:
Proposition 9 A cofibrant object in is precisely a -simplicial set such that is a free -set for each .
The proof is relatively straightforward, and can be found in Goerss-Jardine.
Definition 10 A classifying space for is constructed as follows:
- Take a cofibrant -set which is also a contractible Kan complex.
- Set .
Note that we have required that be cofibrant, which is equivalent to saying that each level is a free -set. It follows from this that is a principal -bundle. Also, we have required that be a contractible Kan complex, as in the classical theory. By the general theory of model categories, we can do this: just take as a cofibrant-fibrant approximation to .
Note as one consequence of this, is itself a Kan complex. This is a special case of the following easy fact: if is a surjective morphism of simplicial sets, with a Kan complex, then is one too.
We now claim that
Theorem 11 (Classification of principal bundles) Let be as above. Then for any simplicial set , pull-back of this bundle induces an isomorphism
Proof: We shall start by proving the following weaker fact. If is any principal -bundle and are simplicially homotopic maps, then the pull-backs are isomorphic as bundles over . One easily reduces to the case . Consider the bundle . Let be the pull-back to . We are going to show that and are isomorphic bundles over ; 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 . The right map, as a bundle map, is a fibration. The lifting thus exists. Since the lifting is a morphism of bundles over , 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 into the set of isomorphism classes of bundles over . We need to show it is both injective and surjective. For the second, let be any bundle; we have to realize it as a pull-back of the bundle . To do this, we note that there is a map of -sets ; we find one such by lifting in the diagram
and using the model structure. Quotienting by gives a map 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 is unique up to equivariant homotopy (again by the model structure) and any equivariant homotopy leads to a homotopy on the quotient.