K-theory of free $G$-sets and the classifying space, and generalization

Question

$\newcommand\Sets{\mathrm{Sets}}\DeclareMathOperator\Nerve{Nerve}$Let $G$ be a finite group, $\mathcal{G}^0$ be the category of finite free $G$-sets and isomorphisms between them. Then $\mathcal{G}^0$ is a symmetric monoidal category with respect to the disjoint union, so we can talk about its $K$-theory, which is homotopy equivalent to $\Sigma ^{\infty}BG_+$.

My first question is, where in the literature can I find this, preferably explicitly stated in this way?

Now, consider the category $\mathcal{G}$ who has unique object and morphisms are elements of $G$. Then we can identify $G$-sets with functors from $\mathcal{G}$ to $\Sets$. Thus the statement above can be reformulated as follows.

$$\Sigma ^{\infty}\lvert\Nerve(\mathcal{G})\rvert_+\simeq K(F^0(\mathcal{G},\Sets)) $$ where $F^0(A,B)$ denote the category whose objects are "free" functors from $\mathcal{G}$ to $\Sets$. And by free, we mean the objects that are in the essential image of the left adjoint of the "forgetful" functor to $\Sets$.

Now my second question is: is there any known sufficient condition on the category $\mathcal{C}$ and its object $C$ so that we have $$\Sigma ^{\infty}\lvert\Nerve(\mathcal{C})\rvert_+\simeq K(F^0(\mathcal{C},\Sets)), $$ where the notation is just as in above except we use the evaluation at $C$ instead of the forgetful functor?

Answer 1

The general point is just that if $\mathcal{U}$ is equivalent to the free symmetric monoidal category $F\mathcal{C}$ generated by $\mathcal{C}$ then $K(\mathcal{U})\simeq\Sigma^\infty_+\mathcal{C}$. Here $F\mathcal{C}$ can be constructed as the category of pairs $(X,C)$, where $X$ is a finite set and $C\in\prod_{x\in X}\text{obj}(\mathcal{C})$. A morphism from $(X,C)$ to $(Y,D)$ consists of a bijection $\sigma\colon X\to Y$ together with a family of $\mathcal{C}$-morphisms $C_x\to D_{\sigma(x)}$ for all $x\in X$. It's not hard to identify the groupoid $\mathcal{F}G$ of finite $G$-sets with the free symmetric monoidal category generated by subgroupoid $\text{Orb}(G)$ of transitive $G$-sets. We can choose subgroups $H_1,\dotsc,H_m$ containing one representative of each conjugacy class, and let $W_k=N_G(H_k)/H_k$ denote the Weyl group, considered as a one-object groupoid. Then $\text{Orb}(G)$ is equivalent to the coproduct of the groupoids $W_k$, so we get $$ K(\mathcal{F}G) \simeq \Sigma^\infty_+ B\text{Orb}(G) \simeq \bigvee_k \Sigma^\infty_+ BW_k $$ This is the tom Dieck splitting. If we restrict attention to the subcategory $\mathcal{F}_1G$ of finite free $G$-sets, we find that this is freely generated by the free orbit $G/1$, whose $G$-equivariant automorphism group is the Weyl group $W1=G$, so we get $K(\mathcal{F}_1G)=\Sigma^\infty_+BG$. There are quite a few interesting results in this vein, but unfortunately I do not think that there is a good source in the literature.

Answer 2

I believe you meant to write $Q(BG_+)$ in the first paragraph of your post, where $Q = \Omega^\infty\Sigma^\infty$. The this result is really a folk theorem and is sometimes called the "Barratt–Priddy–Quillen–Segal" theorem. This is not a folk theorem but rather it is a theorem with a folk authorship in that none of these people wrote down the theorem in this context as far as I know.

One way to deduce it is to use the the Group Completion Theorem (see e.g., McDuff, D.; Segal, G. Homology fibrations and the “group-completion” theorem. Invent. Math. 31 (1975/76), no. 3, 279–284).

The classifying space the category of finite free $G$-sets and their isomorphisms defines a topological monoid $M$. The group completion theorem tells us in this case that $\Omega B M$ coincides with $Q(BG_+)$.

Answer 3

$\DeclareMathOperator\End{End}\newcommand\Set{\mathrm{Set}}\DeclareMathOperator\Nerve{Nerve}\DeclareMathOperator\Fun{Fun}$Here's perhaps a way to understand the general case that Neil mentioned, and a way to apply it to your question at the end.

Suppose you have a category $C$. Then there's a free symmetric monoidal category on $C$, $FC$, which Neil described in his answer.

Now if you're looking at $F^0(C,\Set)$ with respect to some $c\in C$, the category that you get only depends on $B{\End(c)}\subset C$, the full subcategory on $c$ (which is a one-object category, so you can see it as a monoid somehow), and in fact its core-groupoid (which is the only thing that $K$-theory depends on) is exactly $F(B{\End(c)^\times})$, i.e. finite free $\End(c)^\times$-sets, where $\End(c)^\times$ is the subgroup of invertible elements of $\End(c)$.

The reason for this is that left Kan extension along $\{c\}\to C$ factors as left Kan extension along $\{c\}\to B{\End(c)}$, and then left Kan extension along $B{\End(c)}\to C$, but the latter is a full subcategory inclusion, therefore left Kan extension along it is one as well, so if you're looking at the category of people that are left Kan extended from $\{c\}$ (from a finite set I would assume, to avoid Eilenberg swindle type phenomena), you might as well look at the same subcategory, but in $\Fun(B{\End(c)},\Set)$, so you might as well assume $C = B{\End(c)}$. Then you can notice that a good description of this category is just finite copies of $\End(c)$ with its $\End(c)$-action, and the core-groupoid of that will just be the same thing but for $\End(c)^\times$.

Therefore you get exactly the same situation as for a group : $K(\Fun^0(C,\Set)) = \Sigma^\infty_+(B(\End(c)^\times))$ where my $B$ is your $\lvert\Nerve({-})\rvert$.

Now the reason for that thing is that when $C$ is a groupoid (e.g. $B(\End(c)^\times)$), (the nerve of) $FC$ happens to also be the free symmetric monoidal $\infty$-groupoid on $C$, i.e. the free $E_\infty$-space on $\lvert\Nerve(C)\rvert$; hence if you apply group completion to it, you get the free grouplike $E_\infty$-space on it, in other words (up to delooping) the free connective spectrum on it. But that is precisely $\Sigma^\infty_+\lvert\Nerve(C)\rvert$. As Neil pointed out, this gets you the tom Dieck splitting, and in fact with the right setup for $G$-spectra it can give you an "equivariant Barratt–Priddy–Quillen" theorem.

Certainly this idea is present in Segal's Categories and cohomology theories — where he gives a proof "along those lines" of the Barratt–Priddy–Quillen theorem. A modern reference where this idea is explicitly used that way to prove the tom Dieck splitting is Barwick's Spectral Mackey functors and equivariant algebraic $K$-theory (I), specifically theorem A.9. His argument can be generalized to describe the free $E_\infty$-space on a $1$-groupoid.