(This is the second post devoted to unpacking some of the ideas in Segal’s paper “Categories and cohomology theories.” The first is here.)

Earlier, I described an observation (due to Beck) that loop spaces could be characterized as algebras over the monad {\Omega \Sigma}. At least, any loop space was necessarily an algebra over that monad, and conversely any algebra over that monad was homotopy equivalent to a loop space. There is an alternative and compelling idea of Segal which gives a condition somewhat easier to check.

As far as I understand, most of the different approaches to delooping a space consist of imitating the classical construction for a topological group {G}: the construction of the space {BG}. It is known that any topological group {G} is (weakly) homotopy equivalent {\Omega BG}, and conversely (though perhaps it is not as well known) that any loop space is homotopy equivalent to a topological group. (This can be proved using the simplicial construction of Kan.) Given a space (which may not be a topological group), the idea is that delooping machinery will assume given just enough structure to build something analogous to the classifying space, and then build that. This is, for instance, how the construction of Beck ran.

Here’s Segal’s idea; it is quite similar to the {\Gamma}-idea. Given a topological group {G}, we can construct {BG} using a standard simplicial construction. If {G} is only a group object in the homotopy category, we can’t run this construction. Segal decides just to assume that one has given the data of a simplicial object that behaves like {BG } should and runs with that.

The starting point is that one can encode the structure of a monoid in a simplicial set. Given a monoid {G}, the simplicial set {BG} has the following properties.

  1. {(BG)_0} is a point.
  2. The map {(BG)_n \rightarrow \prod_{i=1}^n (BG)_1} induced by the {n} inclusions {[1] \rightarrow [n]} (sending {0} and {1} to consecutive elements) is an isomorphism.

In fact, if we have any simplicial set with the above properties, it determines a unique monoid. This is proved in a similar way. If {X_\bullet} is such a simplicial set, then we take {X_1} as the underlying set of the monoid, and the map {X_1 \times X_1 \rightarrow X} comes from the boundary map {X_2 \rightarrow X_1}; the identity element comes from the map {X_0 = \ast \rightarrow X_1}. So monoids can be described as simplicial sets satisfying certain properties (just as commutative monoids can).

As before, we can weaken this by replacing “isomorphism” by “homotopy equivalence.” (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…)