Restricting spectra to finite $n$-truncated/$n$-connected pointed spaces

Question

$\renewcommand{\S}{\mathcal{S}}\newcommand{\l}{\langle}\newcommand{\r}{\rangle}\newcommand{\op}{\mathsf{op}}\newcommand{\fin}{\mathrm{fin}}$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)

  1. 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.
  2. 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

  1. $E$ is excisive if it sends pushouts to pullbacks.
  2. $E$ is reduced if $E(*)\simeq *$;

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?

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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:

1. 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?
2. 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 $\mathrm{Ran}_{\iota_{2}}$?

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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

1. 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;
2. A $(-1)$-commutative monoid is precisely an $\mathbb{E}_{0}$-monoid, meaning a pointed object;
3. 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.

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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.

Answer 1

$\renewcommand{\S}{\mathcal{S}}\newcommand{\l}{\langle}\newcommand{\r}{\rangle}\newcommand{\op}{\mathsf{op}}\newcommand{\fin}{\mathrm{fin}}$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?

Yes, the easiest way to see this is to observe that the category $Γ^\op$ of finite pointed sets embeds in the category $\S^\fin_*$ by sending a finite pointed set to the corresponding discrete pointed simplicial set.

The restriction functor along this inclusion induces a right Quillen functor from simplicial W-spectra to Γ-spaces. In order to see this, it suffices to show that local objects are preserved, i.e., reduced excisive functors restrict to very special Γ-spaces.

Consider some reduced excisive functor $E$ together with its restriction to finite pointed sets, which we want to show is a very special Γ-space.

A Γ-space is special if it sends finite coproducts to finite homotopy products. Finite coproducts are generated by empty coproducts and binary coproducts. The coproduct of the empty family is the initial pointed set, which is sent to a weakly contractible space precisely if the original functor $E$ is reduced. Excisive functors send binary coproducts to homotopy products by definition.

A Γ-space $E$ is very special if $π_0(E_1)$ is a group. This follows immediately from considering the homotopy pushout diagram for $⟨0⟩←⟨1⟩→⟨0⟩$ (with some apex $A$), which is sent to a homotopy pullback diagram with legs $E_0←E_1→E_0$. Since $E_0$ is contractible, this homotopy pullback diagram exhibits $E_1$ as the loop space of $E(A)$, so $π_0(E_1)$ is a group.