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…)