$\renewcommand{\S}{\mathcal{S}}\newcommand{\l}{\langle}\newcommand{\r}{\rangle}\newcommand{\op}{\mathsf{op}}\newcommand{\fin}{\mathrm{fin}}$Recently I've noticed that the definitions of special $\Gamma$-spaces and spectra are quite close in spirit:

**$\Gamma$-spaces**are pointed functors $X\colon(\Gamma^\op,\l0\r)\to(\mathcal{S},*)$ from Segal's category to the category $\mathcal{S}$ of spaces. Moreover, we call $X$ (see Definition 7.1 here)**special**if, for each $\l n\r,\l m\r\in\mathrm{Obj}(\Gamma)$, the map $$X_{\l n\r\vee\l m\r}\to X_{\l n\r}\times X_{\l m\r}$$ induced by the inert surjections $\l n\r\vee\l m\r\to\l n\r$ and $\l n\r\vee\l m\r\to\l m\r$ is a weak equivalence.**very special**if it is special and equivalently- $\pi_0(X_{\l1\r})$ is a group.
- The map $$X_{\l 2\r}\to X_{\l1\r}\times X_{\l1\r}$$ induced by the total map $\l2\r\to\l1\r$ and one of the two inert surjections $\l2\r\to\l1\r$ is a weak equivalence.

**Spectra**are reduced excisive functors $E\colon\mathcal{S}^\fin_*\to\S$ of $\infty$-categories, where- $E$ is
**excisive**if it sends pushouts to pullbacks. - $E$ is
**reduced**if $E(*)\simeq *$;

- $E$ is

Since very special $\Gamma$-spaces are equivalent to connective spectra, this made me wonder: can we view nonconnectivity as arising from enlarging Segal's category $\Gamma^{\mathsf{op}}\overset{\mathrm{def}}{=}\mathsf{Sk}(\mathsf{FinSets}_*)$ of finite pointed sets into the $\infty$-category $\S^\fin_*$ of finite pointed spaces?

In particular, two natural intermediate steps to consider between $\Gamma^{\mathsf{op}}$ and $\S^\fin_*$ are, for each $n\in\mathbb{N}$, the $\infty$-categories $\S^{\fin}_{\leq n,*}$ and $\S^{\fin}_{>n,*}$ of $n$-truncated and $n$-connected spaces, which come with natural inclusions of $\infty$-categories $\iota_{1},\iota_{2}\colon\S^{\fin}_{\leq n,*},\S^{\fin}_{>n,*}\hookrightarrow\S^\fin_*$.

**Questions:**

- Can we describe functors $E\colon\S^{\fin}_{\leq n,*}\hookrightarrow\S$ or $E\colon\S^{\fin}_{>n,*}\hookrightarrow\S$ in terms of spectra? Are the former maybe somehow related to $(-n-1)$-connected spectra?
- Since $\mathcal{S}$ is co/complete, we can consider left and right Kan extensions along $\iota_{1}$ and $\iota_{2}$, giving functors of the form $$\mathrm{Lan}_{\iota_{1}}\colon\mathsf{Exc}_*(\mathcal{S}^\fin_{\leq n,*})\to\mathsf{Sp}.$$ What are the essential images of $\mathrm{Lan}_{\iota_{1}}$, $\mathrm{Lan}_{\iota_{2}}$, $\mathrm{Ran}_{\iota_{1}}$, and $\mathrm{Ran}_{\iota_{2}}$?

Lastly, let me mention that this was asked also by Jonathan on the Homotopy Theory Discord. There, Rune Haugseng mentioned this paper of Harpaz, where one finds a related construction. Harpaz uses spans of $n$-finite spaces (meaning $n$-truncated spaces which additionally have finite homotopy groups), formulating a notion of $n$-commutative monoid for finite $n$ (which has since been extended to the $n=\infty$ case by Carmeli–Schlank–Yanovski). These are related to $n$-semiadditivity, and extend the commutativity ladder $\mathbb{E}_{1}$, $\mathbb{E}_{2}$, $\ldots$, $\mathbb{E}_{\infty}$ of monoid objects in an $\infty$-category in such a way that

- A $(-2)$-commutative monoid is precisely an $\mathbb{E}_{-1}$-monoid (where $\mathbb{E}_{-1}\overset{\mathrm{def}}{=}\mathsf{Triv}^\otimes$), meaning just an object;
- A $(-1)$-commutative monoid is precisely an $\mathbb{E}_{0}$-monoid, meaning a pointed object;
- A $0$-commutative monoid is precisely an $\mathbb{E}_{\infty}$-monoid.

One could imagine also more generally expanding the definition of $\infty$-operad, replacing $\mathrm{N}_{\bullet}(\mathsf{Fin}_*)$ by $\S^\fin_*$ and variants. For spans in $m$-finite spaces, this should give a notion of $m$-operad, as pointed out by Shachar Carmeli on the homotopy Discord.

**Edit:** This question has been mostly answer by Marc Hoyois and Dmitri Pavlov below (thanks!). The only remaining part is what are reduced excisive functors $\mathcal{S}^{\mathrm{fin}}_{*,\leq n}\to\mathcal{S}$, which I've split into a separate question here.