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

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

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?

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