It's possible to view nonconnectivity for spectra as arising from enlarging Segal's category $\Gamma^\mathsf{op}\overset{\mathrm{def}}{=}\mathsf{Sk}(\mathrm{FinSets}_*)$ to the $\infty$-category of finite pointed spaces $\mathcal{S}^\mathrm{fin}_*$, see Dmitri's answer here.
I'm interested on whether the following refinement of this statement is true:
We have an equivalence of $\infty$-categories $$\mathsf{Exc}_*(\mathcal{S}^{\mathrm{fin}}_{\leq n,*},\mathcal{S})\overset{\mathrm{eq.}}{\cong}\mathsf{Sp}_{\geq-n},$$ where
- $\mathsf{Exc}_*(\mathcal{S}^{\mathrm{fin}}_{\leq n,*},\mathcal{S})$ is the $\infty$-category of reduced excisive functors from the $\infty$-category $\mathcal{S}^{\mathrm{fin}}_{\leq n,*}$ of finite $n$-truncated pointed spaces;
- $\mathsf{Sp}_{\geq-n}$ is the $\infty$-category of $(-n-1)$-connected spectra.
The $n=0$ case of the above statement is true; again, see Dmitri's answer.
Additionally, we have a fully faithful inclusion of $\infty$-categories $\iota\colon\mathcal{S}^{\mathrm{fin}}_{\leq n,*}\hookrightarrow\mathcal{S}^{\mathrm{fin}}_{*}$, giving functors \begin{align*} \iota^* &\colon \mathsf{Sp}\to\mathsf{Exc}_*(\mathcal{S}^{\mathrm{fin}}_{\leq n,*},\mathcal{S}),\\ \mathrm{Lan}_{\iota} &\colon \mathsf{Exc}_*(\mathcal{S}^{\mathrm{fin}}_{\leq n,*},\mathcal{S})\to\mathsf{Sp},\\ \mathrm{Ran}_{\iota} &\colon \mathsf{Exc}_*(\mathcal{S}^{\mathrm{fin}}_{\leq n,*},\mathcal{S})\to\mathsf{Sp}. \end{align*} If we indeed have $\mathsf{Exc}_*(\mathcal{S}^{\mathrm{fin}}_{\leq n,*},\mathcal{S})\overset{\mathrm{eq.}}{\cong}\mathsf{Sp}_{\geq-n}$ as above, it would be natural to wonder if $\iota^*$ is just $(-n-1)$-truncation of spectra. If the above quoted statement is true, is $\iota^*$ indeed given by $(-n-1)$-truncation? What about $\mathrm{Lan}_{\iota}$ and $\mathrm{Ran}_{\iota}$?
