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.

$$\Sigma ^{\infty}|Nerve(\mathcal{G})|_+\simeq K(F^0(\mathcal{G},Sets)) $$ where $F^0(A,B)$ denote the category whose objects are "free" functor 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 $$\Sigma ^{\infty}|Nerve(\mathcal{C})|_+\simeq K(F^0(\mathcal{C},Sets)), $$ where the notation is just as in above except we use the evaluation at $C$ instead of the forgetful functor?

  • $\begingroup$ I thought $K$-theory of a category is the one of its classifying space. But, it seems you are identifying the suspension spectrum of the classifying space as the $K$-theory of your category?! $\endgroup$ – user51223 Jan 16 at 20:23
  • $\begingroup$ c.f. my comment on the answer by John Klein. $\endgroup$ – user43326 Jan 17 at 6:56

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.

  • $\begingroup$ Maybe I am asking something dumb, but how does "The general point" work? $\endgroup$ – user43326 Jan 17 at 7:37

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 $Q(BG_+)$.

  • $\begingroup$ Actually I really meant $BG_+$ because some author use the word $k$-theory to mean the associated spectrum (infinite delooping) and not the infinite loop space. $\endgroup$ – user43326 Jan 17 at 6:48
  • $\begingroup$ @user43326 the K-theory of finite free $G$-sets is not $BG$, it is $Q(BG_+)$. Here is a simple reason why it can't be $BG$: the latter is not in general an infiinite loop space (but the $K$-theory is an infinite loop space). By the way, $G$ does not need to be finite in the Barratt-Priddy-Quillen-Segal theorem. $\endgroup$ – John Klein Jan 17 at 12:14
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    $\begingroup$ that is exactly what is I am saying, for some authors it is the spectrum and not the infinite loop space... $\endgroup$ – user43326 Jan 17 at 15:15
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    $\begingroup$ Oh, thank you, I had missed that. In most of my writing, I identify a based space with its suspension spectrum... Corrected now. $\endgroup$ – user43326 Jan 19 at 8:08
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    $\begingroup$ @JohnKlein different people use different notation. I would quite often leave the $\Sigma^\infty$ implicit in this kind of context. $\endgroup$ – Neil Strickland Jan 19 at 9:18

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