From what I understand, the [Borel construction](https://ncatlab.org/nlab/show/Borel+construction) takes a $G$-space $X$ and produces a topological space $X\times\_{G}\mathbf{E}G$―the homotopy quotient $X/\!\!/G$ of $X$ by $G$ in the $\infty$-category of spaces $\mathcal{S}$―satisfying
$$\mathrm{H}^\*(X\times\_G\mathbf{E}G;R)\cong\mathrm{H}^\*\_G(X;R).$$
One partiularly important example is given by $\mathbf{B}G$, the Borel construction / homotopy quotient of the point: $\mathbf{B}G\simeq\mathbf{E}G\times\_G\*$.


Moving to the pointed setting, one has the [pointed Borel construction](https://mathoverflow.net/questions/105161), which takes a pointed $G$-space $X$ and returns $\mathbf{E}G\_+\wedge\_{G}X$, the homotopy quotient of $X$ by $G$ in the $\infty$-category of pointed spaces $\mathcal{S}\_\*$. Concretely, it is given by
\begin{align\*}
\mathbf{E}G\_+\wedge\_{G}X &\overset{\mathrm{def}}{=} \frac{\mathbf{E}G\times\_G X}{\mathbf{E}G\times\_G\*},\\
&\cong \frac{\mathbf{E}G\times\_G X}{\mathbf{B}G}.
\end{align\*}
Now, $\mathbf{E}G\_+\wedge\_G\*\cong\*$, rather than $\mathbf{B}G$. But while $\*$ is the monoidal unit of $\mathcal{S}$, it is not that of $\mathcal{S}\_\*$, which is $S^{0}$. Hence it would be interesting to know:


**Question.** What is the pointed Borel construction $\mathbf{E}G\_+\wedge\_G S^0$ of the $0$-sphere? Is it related to $\mathbf{B}G\_+$?



What is the pointed Borel construction of the $0$-sphere?
Let's apply your definition (which I think has typos on the RHS - the two "+" subscripts on the EG should not be there I think). Let's model everything as topological spaces and do the calculation there. Let's model $S^0$ as the discrete set $\{p,q\}$ with basepoint $p$.


$$EG\_+\wedge\_GS^0:=\frac{EG\times\_G\{p,q\}}{EG\times\_G\{p\}}$$
$$\simeq \frac{EG\times\_G\{p\}\coprod EG\times\_G\{q\}}{EG\times\_G\{p\}}$$
$$\simeq \frac{BG\coprod BG}{BG}$$
$$\simeq \text{pt} \coprod BG $$
$$\simeq BG\_+$$


