3
$\begingroup$

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?

$\endgroup$
  • $\begingroup$ 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
5
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ Maybe I am asking something dumb, but how does "The general point" work? $\endgroup$ – user43326 Jan 17 at 7:37
2
$\begingroup$

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 $Q(BG_+)$.

$\endgroup$
  • $\begingroup$ 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
  • 1
    $\begingroup$ 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
  • 1
    $\begingroup$ 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
  • 3
    $\begingroup$ @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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.