Apologies for the long silence on this blog! I’ve been distracted for the past month with other things. I’m taking several interesting math classes this semester, one of which I have been liveTeXing and one of which has (at least for now) been providing notes. I’ve also been taking the Kan seminar at MIT. I recently gave a half-hour talk for that class, on Segal’s paper “Categories and cohomology theories;” as a short talk, it focused mostly on the key definitions in the paper rather than the proofs. (Reading Gian-Carlo Rota’s book has convinced me that’s not a bad thing.)  This post contains the notes for that talk.

The goal of this talk is to motivate the notion of “homotopy coherence,” and in particular the example of (special) ${\Gamma}$-spaces. In particular, the goal is to find a “homotopy coherent” substitute for the notion of a topological abelian monoid.

Why might we want such a notion? Given a topological abelian monoid ${A}$, one can form the classifying space ${BA}$, which still acquires the structure of a topological abelian monoid (if we use the usual simplicial construction). It follows that we can iterate the construction, producing a sequence of topological abelian monoids

$\displaystyle A, BA, B^2 A, \dots \ \ \ \ \ (1)$

together with maps ${B^{n-1}A \rightarrow \Omega B^nA}$. If ${A}$ is a topological abelian group, then these maps are all equivalences, and we have an infinite delooping of ${A}$. Therefore, we can extract a cohomology theory from ${[\cdot, A]}$.

If ${A}$ is not assumed to be a group, then the maps ${B^{n-1} A \rightarrow \Omega B^n A}$ are still equivalences for ${n \geq 2}$ and we get a cohomology theory out of ${[\cdot, \Omega BA]}$. This ability to extract an infinite loop space is a very desirable property of topological abelian monoids. Unfortunately, topological abelian monoids are always products of Eilenberg-MacLane spaces, but we’d like to deloop other spaces. Motivated by this, let us take the delooping question as central and declare:

Requirement: A good definition of a “homotopy coherent” topological abelian monoid should have the following properties. If ${X}$ is a homotopy coherent topological monoid, then:

1. ${X}$ should come with the structure of a homotopy commutative ${H}$ space.
2. We should be able to build a sequence of spaces ${X, BX, B^2 X, \dots }$ and maps ${B^{n-1 } X \rightarrow \Omega B^n X}$ such that if ${X}$ is grouplike (i.e., ${\pi_0 X}$ is a group), then these maps are equivalences (so that we get an infinite delooping of ${X}$).
3. Conversely, any infinite loop space should be a candidate for ${X}$.
4. Finally, the definition should not require explicitly assuming a delooping to begin with!

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…)