Let $\mathcal C$ be a permutative category, that is a symmetrical monoidal category with strict associativity. One can then define the $K$-groups of $\mathcal C$, for $n >0$ by
$$K\_n(\mathcal C) = \pi\_n(\Omega B |\mathcal C|),$$
where $|C|$ denotes the realization of the nerve of $\mathcal C$ that inherits a multiplication coming from the monoidal structure.


My question is: If all Hom-sets in $\mathcal C$ are finite, are the groups $K\_n(\mathcal C)$ then also all finite?


Here are there three examples I have seen that motivated this question:


(1) If $\mathcal C$ is given by an abelian group $G$, we get $\Omega B |\mathcal C| = BG$.


(2) If $\mathcal C$ is the category of finite sets with inclusions, we have $\Omega B |\mathcal C| = (B\Sigma\_{\infty})^+ = Q\_0S^0$ by Baratt-Priddy-Quillen, hence we get the stable homotopy groups of spheres.


(3) For $R$ a (commutative) ring, we can take $\mathcal C$ to be free modules of finite rank and get the higher algebraic $K$-groups of $R$. Quillen computed them for $R = \mathbb F\_q$ a finite field, they are $K\_{2i} = 0$ and $K\_{2i-1} = \mathbb Z/(q^i-1)$.


Disclaimer: I have just started to learn about higher K-theory and this question is motivated by my ignorance, so feel free to close it if it's stupid.



K-groups of a permutative category - are they finite?
First, I do not think that strict associativity makes a difference, so I will ignore it.


Next, let $G$ be a finite group, and let $\mathcal{C}G$ be the category of finite $G$-sets and equivariant bijections (which is symmetric monoidal under disjoint union). Then 
$$ K(\mathcal{C}G)=\Omega^\infty\Sigma^\infty\left(\coprod\_{(H)} BW\_GH\right)\_+ 
 = \Omega^\infty (S\_G)^G.
$$
Here $H$ runs over conjugacy classes of subgroups, and $W\_GH$ is $(N\_GH)/H$, where $N\_GH$ is the normaliser of $H$. Also, $S\_G$ is the equivariant sphere spectrum, and $(S\_G)^G$ is the fixed point spectrum in the sense of Lewis and May. This gives 
$$ \pi\_1 K(\mathcal{C}G) = \mathbb{Z}/2\oplus \bigoplus\_{(H)} \pi\_1^S(BW\_GH) = \mathbb{Z}/2\oplus\bigoplus\_{(H)} (W\_GH)\_{\text{ab}}.
$$
(The factor $\mathbb{Z}/2$ is $\pi\_1$ of the sphere spectrum, contributed by the disjoint basepoint.)


This is of course finite. However, we can also consider the category $\mathcal{C}\mathbb{Z}$ of finite sets with an action of $\mathbb{Z}$. Such an action must factor through $\mathbb{Z}/n!$ for some $n$, so $\mathcal{C}\mathbb{Z}$ is the colimit of the sequence of categories $\mathcal{C}\mathbb{Z}/n!$, for which we have
$$ \pi\_1 K(\mathcal{C}\mathbb{Z}/n!) = \mathbb{Z}/2\oplus\bigoplus\_{d|n!} \mathbb{Z}/d. $$
I think it works out that the terms in the colimit assemble in the obvious way to give 
$$ \pi\_1 K(\mathcal{C}\mathbb{Z}) = \mathbb{Z}/2\oplus\bigoplus\_{d>0} \mathbb{Z}/d, $$
and this is infinite, even though all hom sets in $\mathcal{C}\mathbb{Z}$ are finite.


Of course, the monoid of isomorphism classes in $\mathcal{C}\mathbb{Z}$ is infinitely generated. It would be more subtle (or perhaps even impossible?) to find a counterexample without that property?


