(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…)