Functor category with only isomorphisms in the Gamma space construction

Question

I'm looking at the Gamma space construction in the paper "Gamma spaces and information" by Matilde Marcolli. For a pointed set $X$ one takes the category $P(X)$ of pointed subsets of $X$ and inclusions, a category $\mathcal{C}$ with coproducts and a zero, and then looks at the functor category $\Sigma_\mathcal{C}(X)$ of summing functors $P(X) \rightarrow \mathcal{C}$. Summing functors send base point of $X$ to zero of $\mathcal{C}$, $\Phi_X(\ast) = 0$, and for $A \cap B = \{\ast\}$, $\Phi_X(A \cup B) = \Phi_X(A) \oplus \Phi_X(B)$, where $\oplus$ denotes the coproduct of $\mathcal{C}$. The Gamma space construction for $X$ is finalized by taking the nerve of the category of summing functors, $\mathcal{N}\Sigma_\mathcal{C}(X)$.

It is then stated that the morphisms in $\Sigma_\mathcal{C}(X)$ are the natural isomorphisms. My question is why this is done? Is there some, possibly basic reason I'm missing, to take only the isomorphisms as morphisms of your category. In this construction or in general. Looking at the answer Category with one isomorphism class, is this somehow related to enforce skeletality on the category and then the nerve above becomes nicer?

Answer 1

Unfortunately I have not read Marcolli's paper, but it sounds like you are describing a construction that goes back to Segal in ``Categories and cohomology theories''.

I think the short answer to your question is people take the category of isomorphisms because in this way one tends to get the most interesting homotopy type. Suppose for example you took the category of summing functors and all natural transformations between them. Since the category $C$ has has a zero object, it follows that the category of summing functors has an initial object, so the geometric realization of the category of summing functors would be contractible.

Segal's motivation was to construct interesting infinite loop spaces out of categories. The idea (or rather my crude simplification of it!) is that if $C$ is a symmetric monoidal category, then its classifying space $BC$ is not quite an infinite loop space, but it is close: it is an infinite loop space after group completion. The category of summable functors from a set with $n$-elements to $C$ is equivalent to $C^n$. Taken together for all $n\ge 0$, they can be used to construct a delooping (of the group completion) of $BC$.

One way to get a symmetric monoidal category is to start with a category with coproducts and then take its full subcategory of isomorphisms. For some reason it turns out that some of the most interesting infinite loop spaces are obtained in this way: the first space of the sphere spectrum colim$_n \Omega^nS^n$ is obtained from the category of finite sets and bijections. The $K$-theory space $\mathbb Z \times BO$ is obtained from the category of vector spaces and isomorphisms, etc. However, there also are interesting examples of infinite loop spaces that are obtained as the group completion of a symmetric monoidal category where not all morphisms are isomorphisms. Various categories of cobordisms come to mind.