Delooping and the bar construction

The present post is motivated by the following problem:

Problem: Given a pointed space , when is of the homotopy type of a -fold loop space for some ?

One of the basic observations that one can make about a loop space is that admits a homotopy associative multiplication map

Having such an H structure imposes strong restrictions on the homotopy type of ; for instance, it implies that the cohomology ring with coefficients in a field is a graded Hopf algebra. There are strong structure theorems for Hopf algebras, though. For instance, in the finite-dimensional case and in characteristic zero, they are tensor products of exterior algebras, by a theorem of Milnor and Moore. Moreover, for a double loop space , the H space structure is homotopy commutative.

Nonetheless, it is not true that any homotopy associative H space has the homotopy type of a loop space. The problem with mere homotopy associativity is that it asserts that two maps are homotopic; one should instead require that the homotopies be part of the data, and that they satisfy coherence conditions. The machinery of operads was developed to codify these coherence conditions efficiently, and today it seems that one of the powers of higher (at least, ) category theory is the ability to do this in a much more general context.

For this post, I want to try to ignore all this operadic and higher categorical business and explain the essential idea of the delooping construction in May’s “The Geometry of Iterated Loop Spaces”; this relies on some category theory and a little homotopy theory, but the explicit operads play very little role.

 

The commutative cochain problem and the polynomial de Rham functor

Let be a field. The commutative cochain problem over is to assign (contravariantly) functorially, to every simplicial set , a commutative (in the graded sense) -algebra , which is naturally weakly equivalent to the algebra of singular cochains (with -coefficients). We also require that is a surjection whenever . Recall that is an associative algebra, but it is not commutative; the commutativity only appears after one takes cohomology. The commutative cochain problem attempts to find an improvement to .

If has finite characteristic, this problem cannot be solved, owing to the existence of nontrivial cohomology operations. (The answers at this MathOverflow question are relevant here.) However, there is a solution for , given by the polynomial de Rham theory. In this post, I will explain this.

 

The classifying space of a simplicial group

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.

 

Theorem A via general formalism

The following result is useful in algebraic K-theory.

Theorem 1 Let be a functor between categories. Suppose is contractible for each . Then is a weak homotopy equivalence.

I don’t really know enough to give a good justification for the usefulness, but in essence, what Quillen did in the 1970s was to show that the Grothendieck group of an “exact category” could be interpreted homotopically as the fundamental group of the nerve of the “Q-category” built from the exact category. As a result, Quillen was able to define higher K-groups as the higher homotopy groups of this space. He then proved a lot of results that were proved by ad hoc, homological means for the Grothendieck group of a category for the higher K-groups as well, by interpreting them in terms of homotopy theory. This result (together with the extension, “Theorem B”) is a key homotopical tool he used to analyze these nerves.

Here denotes the nerve of the category : it is the simplicial set whose -simplices consist of composable strings of morphisms of . The overcategory has objects consisting of pairs for , a morphism in ; morphisms in are morphisms in making the natural diagram commute. We say that a category is contractible if its nerve is weakly contractible as a simplicial set.

There are other reasons to care. For instance, in higher category theory, the above condition on contractibility of over-categories is the analog of cofinality in ordinary category theory. Anyway, this result is pretty important.

But what I want to explain in this post is that “Theorem A” (and Theorem B, but I’ll defer that) is really purely formal. That is, it can be deduced from some standard and not-too-difficult manipulations with model categories (which weren’t all around when Quillen wrote “Higher algebraic K-theory I”).

To prove this, we shall obtain the following expression for a category:

where ranges over the objects of . This expresses the nerve of as a colimit of simplicial sets arising as the nerves of . We will compare this with a similar expression for the nerve of , that is . Then, the point will be that is a weak equivalence for each ; this by itself does not imply that the induced map on colimits is a weak equivalence, but it will in this case because both the colimits will in fact turn out to be homotopy colimits. I’ll start by explaining what those are.

 

The Dold-Kan correspondence III

Recall that we were in the middle of establishing a crucial equivalence of categories between simplicial abelian groups and chain complexes. Last time, we had defined the two functors: on the one hand, we had the normalized chain complex of a simplicial abelian group; on the other hand, we had defined a functor that amalgamated a chain complex into a simplicial abelian group. We were in the middle of proving that the two functors were quasi-inverse.

With the same notation as before, we were trying to prove:

Proposition 3 (One half of Dold-Kan) For a simplicial abelian group , we have for each , an isomorphism of abelian groups

Here the map is given by sending a summand to via the pull-back by the term . Alternatively, the morphism of simplicial abelian groups

is an isomorphism.

 

The Dold-Kan correspondence II

Recall from last time that we were in the middle of proving the Dold-Kan correspondence, an important equivalence of categories between simplicial abelian groups and chain complexes. We defined three functors last time from simplicial abelian groups: the most obvious was the Moore complex, which just spliced all the components into one big chain complex with the differential the alternating sum of the face maps. But we noted that the functor one uses to construct this equivalence is ultimately either the normalized chain complex or the Moore complex modulo degeneracies.

Today, I’ll show that the two functors from simplicial abelian groups to chain complexes are in fact the same, through a decomposition that next time will let us construct the inverse functor. I’ll also construct the functor in the reverse direction. A minor word of warning: the argument in Goerss-Jardine (which seems to be the main source nowadays for this kind of material) has a small mistake! See their errata. This confused me for quite a while.

From chain complexes to simplicial groups

A priori, the normalized chain complex of a simplicial abelian group looks a lot different from , which a priori has much more structure. Nonetheless, we are going to see that it is possible to recover entirely from this chain complex. A key step in the proof of the Dold-Kan correspondence will be the establishment of the functorial decomposition for any simplicial abelian group

Here the map from a factor corresponding to some to is given by pulling back by . We will establish this below.

 

The Dold-Kan correspondence I

The next couple of posts will cover the Dold-Kan correspondence, which establishes an equivalence of categories between simplicial abelian groups and chain complexes. While this will not be strictly necessary for the introduction of the cotangent complex, it is a sufficiently important fact that it seems worth a digression.

As far as I can tell, the Dold-Kan correspondence is a fairly technical result, and I’m not sure I have any good intuition for why one should expect it to work. But at least one can say the following: given a simplicial abelian group (that is, a contravariant functor from the simplex category to the category of abelian groups), one can form a chain complex in a fairly easy manner: just take the -simplices of the simplicial group as the degree part of the complex, and define the boundary using the alternating sum of the simplicial boundary maps (defined below); this is the classical computation that one does in introductory algebraic topology, of showing that the singular chain complex is indeed a complex.

So it’s natural that you would get a chain complex from a simplicial abelian group. Except, as it turns out, this is the wrong functor for the Dold-Kan correspondence; it is, however, close, being right up to (natural) homotopy.

The other bit of intuition that I’ve heard is the following. Given a topological space , there is a means of obtaining the homology of as the homotopy groups of the infinite symmetric product; this is the so-called Dold-Thom theorem. (See this, for instance.) The Dold-Kan correspondence is in some sense the simplicial analog of this. The infinite symmetric product is much like the abelianization functor from simplicial sets to simplicial abelian groups (that applies the free abelian group pointwise). Now, it will come out of the Dold-Kan correspondence that the so-called “simplicial homotopy groups” of a simplicial abelian group are going to be the same thing as the homology of the associated chain complex. This is a rather loose analogy, and my understanding is that one cannot derive Dold-Thom from Dold-Kan.

 

Simplicial sets II: geometric realization

The semester here finished recently, which means that I will (hopefully!) have more time to update this blog over the next couple of months. I intend to continue the brief series on simplicial methods, which should eventually lead into the cotangent complex soon. I have some other ideas for topics in the near future, but given my recent record at keeping promises, I shall perhaps refrain from divulging the information until I actually have the posts ready!

Recall that last time, we introduced the notion of a simplicial set. As these were presheaves on the category of finite ordered sets (that is, the simplex category), we talked for a while about presheaves in general, on any small category. We did some abstract nonsense and showed in particular that any presheaf is (canonically, in fact!) the colimit of representable presheaves. In our case, that meant that the standard simplexes were enough to generate the entire category of simplicial sets. Today, using this formalism, we are going to see that functors can be defined solely on the standard simplices and thus extended canonically.

Now you might be wondering why the simplex category itself is so special, especially since everything we’ve done so far has been for presheaves on any small category. In homotopy theory, which we haven’t gotten to, simplicial sets have the highly important property of admitting a closed model structure which is Quillen equivalent to the Serre model structure; you might thus wonder which categories of presheaves provide a model for classical homotopy theory in the above sense. I don’t have a complete answer; however, it seems worth mentioning that there is work by Cisinski that does. That is, there is a complete characterization of categories whose presheaf categories admit a model structure Quillen equivalent to spaces. But simplicial sets are (presumably) one of the simplest, and have the advantage of arising in many settings.

1.4. Adjunctions

Let be a category, and a cocomplete category. We are interested in colimit-preserving functors

Here, as before, is the category of presheaves on .

We shall, in this post, write functors out of a presheaf category with a line above them, and functors just defined out of without the line. Functors will be in bold.

 

Simplicial sets I

As planned, I’m going to try to say a few words on the cotangent complex in the following posts. There is a bit of background I’d like to go through first, starting with the theory of simplicial sets. The reason is that the cotangent complex is most naturally thought of using the model structure on simplicial rings, on which it is a non-abelian version of a derived functor.

1.1. The simplex category

Definition 1 Let be the category of finite (nonempty) ordered sets and order-preserving morphisms. The object will denote the set with the usual ordering. Thus is equivalent to the subcategory consisting of the . This is called the simplex category.

There is a functor from to the category of topological spaces. Given , we send it to the standard topological -simplex  that consists of points such that each and . Given a morphism of ordered sets, we define by sending

Here the empty sum is to be regarded as zero.

1.2. Simplicial sets

Definition 2 A simplicial set  is a contravariant functor from to the category of sets. In other words, it is a presheaf on the simplex category. A morphism of simplicial sets is a natural transformation of functors. The class of simplicial sets thus becomes a category . Asimplicial object in a category is a contravariant functor .

We have just seen that the category is equivalent to the subcategory consisting of the . As a result, a simplicial set is given by specifying sets for each , together with maps

for each map in . The set is called the set of -simplices of .