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.

**1. Monads**

The suspension and loop functors on the category of pointed topological spaces (at least, a convenient reformulation thereof) are adjoint. Consequently, the composition is an example of a monad. I found the definition of a monad quite confusing until I learned that a monad is essentially a monoid object in an appropriate monoidal category. Namely, the definition of a monad requires that there be a unit map

and a “composition” map

satisfying associativity and unital relations. If one considers the category of endofunctors with the monoidal structure given by composition of functors, then a monad is literally a monoid object.

An *algebra* over a monad is then a “module” (the term “algebra” is perhaps unfortunate) over the “monoid” ; that is, it is an object together with a “multiplication”

such that the two ways to get from (via -multiplication and , and via and ) are the same, and such that multiplication by the unit corresponds to the identity. At least, that’s how I think of it. (This deserves a post of its own.)

To be precise, we have an action

given by evaluation, and this can be interpreted as an action of the *monoidal* category on the category . Now if is a monad, it is a monoid in , and it makes sense for *it* to act on a space. This is precisely a -algebra.

**2. The classifying space construction**

There are a whole bunch of methods of producing a simplicial object from a monoid. Here are a few.

Given an ordinary group (or, for that matter, monoid) , there is a standard simplicial construction (due to Milnor for general topological groups) of the classifying space . Namely, one considers as a category with one object, and takes the nerve of the category. Explicitly, we have that the -simplices are given by . The degeneracy maps correspond to inserting a , while the face maps correspond to multiplying consecutive elements.

The universal cover of is given by the simplicial set , which is simplicially contractible. Here can be thought of geometrically as an infinite join of with itself. Simplicially, the -simplices are given by . It is the nerve of the groupoid whose objects are the elements and such that consists of the unique element such that . At the level of categories, we can think of this category as the “universal cover” of the groupoid from which was constructed.

So if you write everything out, not much is really happening. The simplicial set associates to a poset the set , and to a map of posets the natural map given by pulling back the coordinates. The group structure of doesn’t actually matter for constructing ; the construction actually makes sense for any set with a given basepoint, and produces a contractible simplicial set. But for a group , one notices that acts both on the left and on the right of the simplicial set (though the left and right actions are not the same).

**3. The bar construction**

More generally, let’s suppose is a group. Suppose acts on the *right* on and on the *left* on . Then we can form a simplicial object as follows. In degree zero, it is . In degree one, it is . In degree , it is . The face and degeneracy maps come from not only group multiplication, but from the group actions on . For instance, given , we have

This is a simplicial set, which we shall denote .

Definition 1is thebar constructionfor , and .

When , this is in fact homotopy equivalent to the constant simplicial set . When , in fact we have constructed the simplicial “path space” to the simplicial set constructed above. One can write down an explicit section and a homotopy as follows. Define

In the other direction, we define

The maps described make sense in every dimension, and they are furthermore easily seen to be simplicial maps. Clearly , and it is also possible to show that is homotopic to the identity by writing down an explicit simplicial homotopy, which does not seem terribly enlightening.

**4. Classifying spaces for monads**

Let’s now suppose is a monad, so that it is a monoid in . Then we can say that a functor is a **left module** over if there is a morphism

satisfying the usual module conditions (in a monoidal category). Given an object together with a -algebra structure on (really, a “left module” structure), we can form the **bar construction**

This is a simplicial topological space. In dimension , we have as the space in question; the face and degeneracy maps are the same.

As before, we have:

Proposition 2If , then is homotopy equivalent (in the category of simplicial topological spaces) to .

This is proved using the same universal formulas sketched previously in the case where we were just working with monoids and sets.

**5. The delooping apparatus**

The strategy is to use these bar constructions to “deloop” a space provided with sufficient construction. This is not really all that surprising, because the way of “delooping” a topological group is to form its classifying space .

Here is the main result:

Theorem 3A topological space admitting an action of the monad is homotopy equivalent to a -fold loop space.

In fact, the key idea is that the structure of a monad action is precisely what we need to form the bar construction. We have:

Here both sides are simplicial topological spaces; is identified with the constant simplicial space. Now, all we do is pull out the to write

Next, take geometric realizations. Since as simplicial spaces,

we can take geometric realizations to get

Thus, the -fold delooping of is precisely !

Except that there are a few things to check. First, we should really check that is actually a right module over . That’s pretty easy. Whenever we have a left adjoint and a right adjoint , the composite is a monad, and is a right module over the monad. The map

is given by , where the counit of the adjunction is used. So the application of the bar construction here is actually reasonable.

What’s less obvious is that iterated loop functor (which was applied pointwise in the category of simplicial topological spaces) commutes with geometric realization. This is true under appropriate hypotheses, though the proof in May’s book somewhat technical and I don’t understand it well; it doesn’t seem to be that important to the main ideas, though. A relevant result in the simplicial context seems to be the Bousfield-Friedlander theorem as in Goerss-Jardine.

**6. Other constructions**

While this is a sensible delooping construction, the monad is very complicated. The use of operads has given many much simpler monads, which can more easily be shown to act on a space. In May’s book, the little cubes operad is used. I won’t go into details here, but ultimately May shows that the monad associated to this operad has the property that there is a morphism of monads

which is always a weak equivalence when applied to a connected space. As a result, when is connected and is a -algebra, then one can, as before, form the bar construction

(where is given the right -module structure from ), and this is a -fold delooping of .

A special case of May’s results is the following, when , the space is homotopy equivalent to the James construction on , i.e. the free topological monoid on , for connected.

January 12, 2012 at 5:06 pm

Nice post! I have some vague questions: What is “the domain” of the functor B? As you said, BG makes sense for discrete monoids, abstract groups, a monad via B(pt,G,pt)? , a loop space ??…

Are B and \Omega adjoint functors in a sense? Is \Omega o B homotopic to the identity in a sense?

January 12, 2012 at 5:34 pm

I am sorry, you answered most of these questions already in your post: A loop space is an algebra over the monad T:=\Omega S, so we can define functors between topological spaces and T-algebras by sending a space to its loop space and sending an T-algebra G to BG:=B(S, T, G). Also you explained that \Omega o B is homotopic to the identitiy.

So what about adjointness of $B$ and $\Omega$?

January 12, 2012 at 8:05 pm

The classifying space functor can be defined on any space with a coherently associative multiplication law (e.g. a space with an action of an operad): essentially most delooping machinery is about providing enough data to be able to build such a functor . I don’t know what’s the right statement about adjunctions, but at least when you work with simplicial groups, then the classifying space functor from simplicial groups to pointed simplicial sets is in fact a Quillen equivalence (the adjoint is a model for the loop space which is in fact a simplicial group).

January 13, 2012 at 5:06 am

Thanks!

February 6, 2012 at 2:23 pm

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