Let ${X}$ be a connected pointed space. The spaces ${\Omega^n \Sigma^n X}$ are very large and complicated, but a fairly concrete model for their homotopy type can be given using the theory of operads. As a toy example, let’s take ${n = 1}$. In this case, the James construction gives a homotopy equivalence $\displaystyle JX \rightarrow \Omega \Sigma X,$

where ${JX}$ is the free topological monoid on ${X}$, subject to the relation that the basepoint ${\ast \in X}$ be the identity. In other words, one can describe ${JX}$ by taking the disjoint union ${\bigsqcup_{n \geq 0} X^n}$ (the free topological monoid on ${X}$) and making identifications to make ${\ast}$ be the identity. The space ${JX}$ comes with a canonical filtration $\displaystyle \ast \subset X \subset J_2 X \subset J_3 X \subset \dots \subset JX,$

where ${J_n X}$ denotes elements of ${JX}$ which can be expressed as products of ${\leq n}$ elements of ${X}$. The “associated graded” quotients of ${JX}$ are given by the smash powers ${X^{\wedge n}, n > 0}$.

More generally, let ${\mathcal{O}^{(n)}}$ be the little ${n}$-cubes operad; then ${\mathcal{O}^{(n)}}$ comes with a canonical action on any ${n}$-fold loop space. For a connected space ${X}$, it is a theorem of May (related to delooping machinery) that ${\Omega^n \Sigma^n X}$ is homotopy equivalent to the free ${\mathcal{O}^{(n)}}$-algebra on ${X}$, again subject to the condition that the base-point be the unit element. In other words, we can model ${\Omega^n \Sigma^n X}$ via the space $\displaystyle C_n X \stackrel{\mathrm{def}}{=} \left( \bigsqcup_{j \geq 0} \mathcal{O}^{(n)}(j) \times_{\Sigma_j} X^j \right)/\sim,$

where the relation ${\sim}$ is precisely the one that makes ${\ast}$ the identity element. This space, too, has a filtration via the subspaces $\displaystyle \ast \subset C_n^{1}(X) \subset C_n^{2}(X) \subset \dots,$

where ${C_n^{k}(X)}$ consists of the image of the first ${k+1}$ pieces of the disjoint union above. The associated graded pieces are given by the quotients $\displaystyle \mathrm{gr}_j C_n(X) = (\mathcal{O}^{(n)}(j) \times_{\Sigma_j} X^{\wedge j})/( \mathcal{O}^{(n)}(j) \times_{\Sigma_j} \ast), \quad j > 0:$

in other words, we take the coinvariants of ${\mathcal{O}^{(n)}(j) \times X^{\wedge j}}$ in the category of pointed spaces (rather than in spaces). It turns out that this filtration actually becomes much nicer stably.

Theorem 1 (Snaith) The above filtration splits after taking suspension spectra. That is, we have an equivalence $\displaystyle \Sigma^\infty \Omega^n \Sigma^n X = \bigvee_{j \geq 1} \Sigma^\infty \mathrm{gr}_j C_n(X).$ (more…)

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. (more…)

The idea of an operad was born in an attempt to understand when a space has the homotopy type of a loop space, or more generally an ${n}$-fold loop space. An operad ${\mathcal{O}}$ is supposed to be a collection of operations with different arities with rules about how to compose them. To give an algebra over an operad ${\mathcal{O}}$ is to give a space ${X}$ together with an interpretation of each of these “operations” as actual operations on ${X}$.

Motivation

The concrete motivation is as follows. Consider a pointed space ${(X, \ast)}$, the loop space ${\Omega X}$ is something that you want to think of as approximating a topological monoid. This is because you can compose paths. But the composition is only associative up to homotopy. However, essentially any way of composing a collection of ${k}$ paths is equivalent to another way (i.e., differently parenthesized) of composing that collection of ${k}$ paths, in that order, and that equivalence is canonical (up to homotopy, at least).

This is a somewhat long-winded way of saying that there is a multiplication law on ${\Omega X}$ which is not only homotopy associative, but homotopy coherently associative. What does that mean?

The best analogy I can come up with is from ordinary category theory. Consider the definition of a monoidal category. In a monoidal category ${(\mathcal{C}, \otimes)}$, one usually does not want to require an equality of functors ${X \otimes (Y \otimes Z) = (X \otimes Y) \otimes Z}$; this is considered “evil.” Also, this does not generally happen in practice. Rather, one wants a canonical isomorphism between the functors $\displaystyle X \otimes (Y \otimes Z) \simeq (X \otimes Y) \otimes Z.$ (more…)