commutative monoid is a set ${A}$ together with a multiplication map ${m: A \times A \rightarrow A}$ and a distinguished unit element ${e \in A}$, satisfying certain identities. Let us say that we are interested in a homotopical version of this idea, especially a version of the idea of an abelian group. Then, we could try to work in the category of topological abelian groups, but this is somewhat uninteresting from the point of view of homotopy theory: every topological abelian is weakly homotopy equivalent to a product of Eilenberg-MacLane spaces. Alternatively, we could demand that one has a topological space together with a multiplication law which is commutative up to homotopy; however, as we’ve seen, this isn’t enough structure to perform a construction such as the classifying space.

Segal’s idea, in his paper “Categories and cohomology theories,” is to rephrase the definition of a commutative monoid in such a way as to require only a bunch of sets and maps with them, such that certain ones are isomorphisms. This will lead to an immediate homotopical generalization: replace “isomorphism” with “weak equivalence.”

1. Segal’s category of finite sets

We can define the category (due to Segal) ${\mathcal{F}in_*}$ of finite sets and partially defined maps as follows. The objects of ${\mathcal{F}in_*}$ are finite sets. A morphism ${A \rightarrow B}$ in ${\mathcal{F}in_*}$ is the data of a subset ${A' \subset A}$ and a map ${A' \rightarrow B}$. The composition of two partially defined maps is just the ordinary composition, defined wherever it makes sense.

Suppose ${A}$ is a commutative monoid. We can package the data of ${A}$ into a functor $\displaystyle \widetilde{A}: \mathcal{F}in_* \rightarrow \mathrm{Set}$

by sending a finite set ${S}$ to ${A^S}$. Given a partially defined map ${\theta}$ between ${S}$ and ${S'}$, we get a map ${A^S \rightarrow A^{S'}}$ sending a tuple ${(x_s)}$ to the following tuple: $\displaystyle \theta(x)_{s'} = \sum_{s \in \theta^{-1}(s)} x_s.$

Thus, to each commutative monoid, we can associate a functor ${\widetilde{A}: \mathcal{F}in_* \rightarrow \mathrm{Set}}$. The functors ${\widetilde{A}}$ have the following properties:

1. ${\widetilde{A} (\emptyset) = \ast}$.
2. For each ${n}$, ${\widetilde{A}(n) \rightarrow \prod_{i=1}^n \widetilde{A}(1)}$ is an isomorphism, where the maps are induced by the maps ${\langle n\rangle \rightarrow \langle 1\rangle}$ defined only on ${i}$.

In fact, these two properties are enough to recover ${A}$ and its abelian group structure. Let us be a bit more systematic. We let ${\theta_i: \langle n\rangle \rightarrow \langle 1\rangle}$ be the maps listed above; they are very simple, being defined only at one point (that is, ${i}$). Let us suppose we have a functor ${F: \mathcal{F}in_* \rightarrow \mathrm{Set}}$ such that ${F(\emptyset) = \ast}$ and such that for each ${n}$, the product map $\displaystyle F(\langle n\rangle) \stackrel{\prod \theta_i}{\rightarrow} \prod_i F(\langle 1\rangle)$

is an isomorphism. Then there is a canonically determined abelian monoid structure on ${F(\langle 1\rangle)}$, and one can phrase this as an equivalence of categories between such functors and abelian monoids. (more…)