The present post is motivated by the following problem:

Problem: Given a pointed space ${X}$, when is ${X}$ of the homotopy type of a ${k}$-fold loop space ${\Omega^k Y}$ for some ${Y}$?

One of the basic observations that one can make about a loop space ${\Omega Y}$ is that admits a homotopy associative multiplication map

$\displaystyle m: \Omega Y \times \Omega Y \rightarrow \Omega Y.$

Having such an H structure imposes strong restrictions on the homotopy type of ${\Omega Y}$; for instance, it implies that the cohomology ring ${H^*(\Omega Y; k)}$ 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 ${\Omega^2 Y}$, 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, ${(\infty, 1)}$) 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 suspension ${\Sigma}$ and loop ${\Omega}$ functors on the category of pointed topological spaces (at least, a convenient reformulation thereof) are adjoint. Consequently, the composition ${T = \Omega \Sigma}$ 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

$\displaystyle I \rightarrow T$

and a “composition” map

$\displaystyle T^2 \rightarrow T$

satisfying associativity and unital relations. If one considers the category of endofunctors ${\mathrm{Fun}(\mathcal{T}op, \mathcal{T}op)}$ with the monoidal structure given by composition of functors, then a monad is literally a monoid object.

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

$\displaystyle m: TX \rightarrow X$

such that the two ways to get from ${T^2 X \rightarrow X}$ (via ${T}$-multiplication and ${m}$, and via ${m}$ and ${m}$) 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

$\displaystyle \mathrm{Fun}(\mathcal{T}op, \mathcal{T}op) \times \mathcal{T}op \rightarrow \mathcal{T}op$

given by evaluation, and this can be interpreted as an action of the monoidal category on the category ${\mathcal{T}op}$. Now if ${T}$ is a monad, it is a monoid in ${\mathrm{Fun}(\mathcal{T}op, \mathcal{T}op)}$, and it makes sense for it to act on a space. This is precisely a ${T}$-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) ${G}$, there is a standard simplicial construction (due to Milnor for general topological groups) of the classifying space ${BG}$. Namely, one considers ${G}$ as a category with one object, and takes the nerve of the category. Explicitly, we have that the ${n}$-simplices ${(BG)_n}$ are given by ${G^n}$. The degeneracy maps correspond to inserting a ${\ast}$, while the face maps correspond to multiplying consecutive elements.

The universal cover of ${BG}$ is given by the simplicial set ${EG}$, which is simplicially contractible. Here ${EG}$ can be thought of geometrically as an infinite join of ${G}$ with itself. Simplicially, the ${n}$-simplices are given by ${G^{n+1}}$. It is the nerve of the groupoid whose objects are the elements ${g \in G}$ and such that ${\hom(g, h) }$ consists of the unique element ${s \in G}$ such that ${gs = h}$. At the level of categories, we can think of this category as the “universal cover” of the groupoid from which ${BG}$ was constructed.

So if you write everything out, not much is really happening. The simplicial set ${EG}$ associates to a poset ${[n]}$ the set ${G^{[n]} = G^{n+1}}$, and to a map of posets ${[m] \rightarrow [n]}$ the natural map ${G^{[n]} \rightarrow G^{[m]}}$ given by pulling back the coordinates. The group structure of ${G}$ doesn’t actually matter for constructing ${EG}$; the construction actually makes sense for any set ${X}$ with a given basepoint, and produces a contractible simplicial set. But for a group ${G}$, one notices that ${G}$ acts both on the left and on the right of the simplicial set ${EG}$ (though the left and right actions are not the same).

3. The bar construction

More generally, let’s suppose ${G}$ is a group. Suppose ${G}$ acts on the right on ${X}$ and on the left on ${Y}$. Then we can form a simplicial object as follows. In degree zero, it is ${X \times Y}$. In degree one, it is ${X \times G \times Y}$. In degree ${n}$, it is ${X \times G^n \times Y}$. The face and degeneracy maps come from not only group multiplication, but from the group actions on ${X, Y}$. For instance, given ${(x, g, y)}$, we have

$\displaystyle d_0(x, g, y) = (xg, y), \quad d_1(x, g, y) = (x, gy).$

This is a simplicial set, which we shall denote ${B(X, G, Y)}$.

Definition 1 ${B(X, G, Y)}$ is the bar construction for ${X, G}$, and ${Y}$.

When ${X = G}$, this is in fact homotopy equivalent to the constant simplicial set ${Y}$. When ${Y = \ast}$, in fact we have constructed the simplicial “path space” to the simplicial set ${BG}$ constructed above. One can write down an explicit section ${Y \rightarrow B(G, G, Y)}$ and a homotopy as follows. Define

$\displaystyle s: Y \rightarrow B(G, G, Y) , \quad y \mapsto (e, e, \dots, e, y).$

In the other direction, we define

$\displaystyle p: B(G, G, Y) \rightarrow Y, \quad (g_1, \dots, g_{n+1} , y) \mapsto g_1 \dots g_{n+1}y.$

The maps described make sense in every dimension, and they are furthermore easily seen to be simplicial maps. Clearly ${ps = 1}$, and it is also possible to show that ${sp }$ is homotopic to the identity by writing down an explicit simplicial homotopy, which does not seem terribly enlightening.

Let’s now suppose ${T}$ is a monad, so that it is a monoid in ${\mathrm{Fun}(\mathcal{T}op, \mathcal{T}op)}$. Then we can say that a functor ${S \in \mathrm{Fun}(\mathcal{T}op, \mathcal{T}op)}$ is a left module over ${T}$ if there is a morphism

$\displaystyle S T \rightarrow S$

satisfying the usual module conditions (in a monoidal category). Given an object ${X \in \mathcal{T}op}$ together with a ${T}$-algebra structure on ${X}$ (really, a “left module” structure), we can form the bar construction

$\displaystyle B(S, T, X).$

This is a simplicial topological space. In dimension ${n}$, we have ${ST^n X}$ as the space in question; the face and degeneracy maps are the same.

As before, we have:

Proposition 2 If ${S = T}$, then ${B(T, T, X)}$ is homotopy equivalent (in the category of simplicial topological spaces) to ${X}$.

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 ${B(S, T, X)}$ to “deloop” a space provided with sufficient construction. This is not really all that surprising, because the way of “delooping” a topological group ${G}$ is to form its classifying space ${BG}$.

Here is the main result:

Theorem 3 A topological space admitting an action of the monad ${\Omega^k S^k}$ is homotopy equivalent to a ${k}$-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:

$\displaystyle B(\Omega^k S^k, \Omega^k S^k, X) \simeq X .$

Here both sides are simplicial topological spaces; ${X}$ is identified with the constant simplicial space. Now, all we do is pull out the ${\Omega^k}$ to write

$\displaystyle B(\Omega^k S^k, \Omega^k S^k, X) = \Omega^k B(S^k, \Omega^k S^k, X) .$

Next, take geometric realizations. Since as simplicial spaces,

$\displaystyle X \simeq \Omega^k B(S^k, \Omega^k S^k, X) ,$

we can take geometric realizations to get

$\displaystyle X \simeq \Omega^k |B(S^k, \Omega^k S^k, X)|.$

Thus, the ${k}$-fold delooping of ${X}$ is precisely ${B(S^k, \Omega^k S^k, X)}$!

Except that there are a few things to check. First, we should really check that ${S^k}$ is actually a right module over ${\Omega^k S^k}$. That’s pretty easy. Whenever we have a left adjoint ${L}$ and a right adjoint ${R}$, the composite ${RL}$ is a monad, and ${L}$ is a right module over the monad. The map

$\displaystyle L(RL) \rightarrow L$

is given by ${L(RL) = (LR)L \rightarrow (\mathrm{Id})L = L}$, 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 ${\Omega^k}$ (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 ${\Omega^n S^n}$ 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 ${T}$ associated to this operad has the property that there is a morphism of monads

$\displaystyle T \rightarrow \Omega^n S^n$

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

$\displaystyle B(S^n, T, X)$

(where ${S^n}$ is given the right ${T}$-module structure from ${T \rightarrow \Omega^nS^n}$), and this is a ${k}$-fold delooping of ${X}$.

A special case of May’s results is the following, when ${n = 1}$, the space ${\Omega S X}$ is homotopy equivalent to the James construction on ${X}$, i.e. the free topological monoid on ${X}$, for ${X}$ connected.