Categories and cohomology theories

A 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:

${\widetilde{A} (\emptyset) = \ast}$ .
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.

Let’s see how we get an abelian monoid structure on ${F(\langle 1\rangle)}$ if ${F}$ is a functor with the above two properties. In fact, the monoid structure comes from the everywhere defined map ${m: \langle 2\rangle \rightarrow \langle 1\rangle}$ , which induces a map

$\displaystyle F(\langle 1\rangle)^2 \rightarrow F(\langle 2\rangle) \stackrel{F(m)}{\rightarrow} F(\langle 1\rangle)$

where the first map comes from the postulated isomorphisms. Similarly, one gets a unit element from the map ${F(\emptyset) = \ast \rightarrow F(\langle 1\rangle)}$ from the unique map ${\emptyset \rightarrow \langle 1\rangle}$ in the category ${\mathcal{F}in_*}$ . It takes a bit of work (of chasing through the diagrams to see that one has associativity, etc.), but one can check that one in fact has a commutative monoid.

More generally, in an arbitrary category with products ${\mathcal{C}}$ , we might define a strong ${\Gamma}$ -object to be a functor

$\displaystyle F: \mathcal{F}in_* \rightarrow \mathcal{C}$

with the above properties. Then we find that there is an equivalence of categories between ${\Gamma}$ -objects and abelian monoid objects in ${\mathcal{C}}$ , as before.

Segal’s insight is that if we take the category of topological spaces (or simplicial sets), and replace “isomorphism” in the above with “weak homotopy equivalence,” then we get a much more flexible notion than that of a topological commutative monoid that still has a lot of structure for homotopy theory.

Definition 1 A ${\Gamma}$ -space is a functor ${F: \mathcal{F}in_* \rightarrow \mathrm{Spaces}}$ satisfying the two conditions:

${F(\emptyset) }$ is contractible.

For each ${n}$ , the map
$\displaystyle F(\langle n\rangle) \rightarrow \prod_{i=1}^n F(\langle 1\rangle)$

as above, is a weak homotopy equivalence.

2. ${\Gamma}$ -categories

We might also try using this definition in category theory. A commutative monoid object in category theory is a strict symmetric monoidal category. These are pretty restrictive. For instance, while any monoidal category is equivalent to a strict monoidal one, the same is not true in the symmetric case.

Definition 2 A ${\Gamma}$ -category is a functor ${F: \mathcal{F}in_* \rightarrow \mathrm{Cat}}$ satisfying the two conditions:

${F(\emptyset)}$ is a contractible groupoid.

For each ${n}$ , the map
$\displaystyle F(\langle n\rangle) \rightarrow \prod_{i=1}^n F(\langle 1\rangle)$

as above, is an equivalence of categories.

The claim is that a ${\Gamma}$ -category is more or less a symmetric monoidal category! Given a symmetric monoidal category ${(\mathcal{C}, \otimes, 1)}$ , we can construct a ${\Gamma}$ -category as follows. The functor ${F}$ sends a finite set ${A}$ to ${\mathcal{C}^{A}}$ , and given a partially defined map ${f: A \rightarrow B}$ , we get a functor

$\displaystyle \prod_A \mathcal{C} \rightarrow \prod_B \mathcal{C}, \quad (x_a)_{a \in A} \mapsto (y_b)_{b \in B},$

where

$\displaystyle y_b = \bigotimes_{a \in f^{-1}(b)} x_a.$

Since we are working with a symmetric monoidal category, this is well-defined up to unique isomorphism. Except one should be a bit careful about compositions; ${F}$ might not be strictly functorial but rather 2-functorial (reflecting the fact that the tensor product is not strictly commutative or associative, but only up to natural isomorphism). I think these considerations are why a better way of phrasing the definition is to define a ${\Gamma}$ -category as a cofibered category over ${\mathcal{F}in_*}$ ; by the Grothendieck construction, this is approximately the same thing.

Proposition 3 A symmetric monoidal category ${(\mathcal{C}, \otimes , 1)}$ is equivalent to the following data: A category ${\mathcal{C}^{\otimes}}$ together with a map

$\displaystyle \mathcal{C}^{\otimes} \rightarrow \mathcal{F}in_*$

which makes ${\mathcal{C}^{\otimes}}$ a cofibered category over ${\mathcal{F}in_*}$ , such that the induced maps

$\displaystyle \mathcal{C}^{\otimes}_{\left \langle n\right\rangle} \rightarrow \prod_{i=1}^n \mathcal{C}^{\otimes}_{\left \langle 1\right\rangle}$

as before are equivalences.

Here ${\mathcal{C}^{\otimes}_{\left \langle n\right\rangle}}$ is the fiber of ${\mathcal{C}^{\otimes}}$ over ${\left \langle n\right\rangle}$ . This is ultimately what leads to the definition of a symmetric monoidal ${(\infty, 1)}$ -category, but I don’t really understand enough about ${(\infty, 1)}$ -categories to say much more. For the purposes of this paper, the slightly more “rigid” notion of a ${\Gamma}$ -category (i.e. where we have a functor rather than a pseudofunctor into the category of categories) is enough.

3. From ${\Gamma}$ -categories to ${\Gamma}$ -spaces

Note in particular that the nerve of a ${\Gamma}$ -category is a ${\Gamma}$ -space; this is a consequence of the observation that an equivalence of categories induces a homotopy equivalence on the nerves (though the converse is false).

Here is a slightly different approach.

Example: Let ${\mathcal{C}}$ be a category with finite coproducts. The claim is that we can obtain a ${\Gamma}$ -space out of ${\mathcal{C}}$ , though not quite by taking the nerve (which will be contractible as ${\mathcal{C}}$ has an initial object). For a finite set ${A}$ , let ${\mathcal{P}(A)}$ denote the poset of subsets of ${A}$ , and let ${F_{\mathcal{C}}(A)}$ be the category of functors

$\displaystyle \mathcal{P}(A) \rightarrow \mathcal{C}$

which preserve coproducts, and where the morphisms are natural isomorphisms. So, for instance, an element of ${F_{\mathcal{C}}(\left \langle 2\right\rangle)}$ is the data of three objects ${a_1, a_2, a \in \mathcal{C}}$ together with morphisms

$\displaystyle a_1 \rightarrow a, \quad a_2 \rightarrow a$

which exhibit ${a}$ as a coproduct of ${a_1, a_2}$ .

The claim is that ${F_{\mathcal{C}}(\cdot)}$ becomes now a ${\Gamma}$ -functor. In fact, a morphism ${A \rightarrow B}$ in ${\mathcal{F}in_*}$ is precisely the same thing as a coproduct-preserving functor ${\mathcal{P}(B) \rightarrow \mathcal{P}(A)}$ , so it is clear that ${F_{\mathcal{C}}}$ does in fact define a functor

$\displaystyle F_{\mathcal{C}}: \mathcal{F}in_* \rightarrow \mathrm{Cat}.$

Now we have to check that this satisfies the additional conditions. Clearly ${F_{\mathcal{C}}(\emptyset)}$ is a contractible groupoid (it’s the category of initial objects of ${\mathcal{C}}$ ). Meanwhile there is an equivalence of ${F_{\mathcal{C}}(A)}$ with ${\mathcal{C}^A}$ in an obvious sense, since any coproduct-preserving functor ${\mathcal{P}(A) \rightarrow \mathcal{C}}$ is uniquely determined up to unique isomorphism by where it sends each of the singleton sets. If we apply the nerve functor to ${F_{\mathcal{C}}}$ , we can get a ${\Gamma}$ -space.

Example: Here is an example of the previous idea. Let us take ${\mathcal{C}}$ simply to be the category of finite sets (not ${\mathcal{F}in_*}$ !). We then get a ${\Gamma}$ -space from it. The first term is

$\displaystyle \bigsqcup_{n \geq 0} B\Sigma_n$

since the first term is just the groupoid associated to ${\mathcal{C}}$ , which is this.

Example: We can do the same if we replace ordinary categories by topological categories, so that their nerves are now simplicial spaces (whose geometric realizations we can take to get ordinary spaces). For instance, one can consider the category of finite-dimensional complex vector spaces with a hermitian inner product and isometric imbeddings. This clearly has coproducts, and consequently we get a ${\Gamma}$ -space. The value on ${\left \langle 1\right\rangle}$ is the nerve of the topological category of such vector spaces and isometries between them, which is equivalent to is skeleton and the nerve is thus

$\displaystyle \bigsqcup_{n \geq 0} BU(n).$

So this, too, is the first element in a ${\Gamma}$ -space.

Let ${F}$ be a ${\Gamma}$ -space. When we pass from the category of spaces to the homotopy category, we are applying a monoidal functor. This means that we get a “strong ${\Gamma}$ -object” in the homotopy category, in particular a commutative monoid in the homotopy category. Ultimately, we are going to want to think of the first term of ${\Gamma}$ -spaces as somewhat like infinite loop spaces, except that they’re not quite so because the monoid structure in the homotopy category is not necessarily a group structure. Applying a process called “group completion,” though, we will get honest loop spaces.

There is a theorem (called the “group completion theorem”) which enables one to compute what the group completion is in the cases of ${\bigsqcup_{n} B \Sigma_n}$ and ${\bigsqcup_{n} B U(n)}$ ; see this article. In the first case, it is the space obtained by applying the “plus” construction to ${B\Sigma_\infty}$ ; in the second case, it is simply ${BU = \bigcup BU(n)}$ . In particular, both of these should be infinite loop spaces. In both cases, there are specific reasons for it. First, ${BU}$ is an infinite loop space by Bott periodicity, and second, ${(B\Sigma_\infty)^+}$ is the infinite loop space of the sphere spectrum (that is, ${\varinjlim \Omega^n S^n}$ ) by a theorem of Barratt, Priddy, and Quillen.