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

$$\newcommand\Sets{\mathrm{Sets}}\DeclareMathOperator\Nerve{Nerve}$$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?

• 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?! Jan 16, 2020 at 20:23
• c.f. my comment on the answer by John Klein. Jan 17, 2020 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.

• Maybe I am asking something dumb, but how does "The general point" work? Jan 17, 2020 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 $$Q(BG_+)$$.

• 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. Jan 17, 2020 at 6:48
• @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. Jan 17, 2020 at 12:14
• that is exactly what is I am saying, for some authors it is the spectrum and not the infinite loop space... Jan 17, 2020 at 15:15
• Oh, thank you, I had missed that. In most of my writing, I identify a based space with its suspension spectrum... Corrected now. Jan 19, 2020 at 8:08
• @JohnKlein different people use different notation. I would quite often leave the $\Sigma^\infty$ implicit in this kind of context. Jan 19, 2020 at 9:18

$$\DeclareMathOperator\End{End}\newcommand\Set{\mathrm{Set}}\DeclareMathOperator\Nerve{Nerve}\DeclareMathOperator\Fun{Fun}$$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.