What is a spectrum object in $\infty$-topoi?

Following up on the answer of Simon Henry, let us prove the following statement. For a pro-space $\hat{X} = \{X_i\}_{i \in I}$, we let $Spaces_{/\hat{X}}$ denote the $\infty$-topos defined as the (cofiltered) limit in $Topoi$ of the $I$-family of étale topoi $Spaces_{/X_i}$. We will refer to such $\infty$-topoi as pro-étale $\infty$-topoi.

Claim: Suppose that ${\cal X}$ is an $\infty$-topos which is a loop object in $Topoi$. Then ${\cal X}$ is a left exact localization of a pro-étale $\infty$-topos $Spaces_{/\hat{X}}$ for $\hat{X} \in Pro(Spaces)$. If ${\cal X}$ is a double loop object then ${\cal X}$ itself is a pro-étale $\infty$-topos. In particular, every spectrum object in $Topoi$ consists of pro-étale $\infty$-topoi.

Proof: Let ${\cal Y}$ be an $\infty$-topos equipped with two points $x_*,y_*: Spaces \to {\cal Y}$. Before considering the associated limit $Spaces \times_{\cal Y} Spaces$ we can consider the corresponding lax limit (or comma object) $Spaces \times^{\rm lax}_{\cal Y} Spaces$. We claim that this comma object exists and is furthermore a pro-étale $\infty$-topos. Indeed, let ${\cal Z}$ be an $\infty$-topos and let $p_*: {\cal Z} \to Spaces$ denote the terminal map. Then the data of a natural transformation $x_*p_* \Rightarrow y_*p_*$ is equivalent, by adjunction, to the data of a natural transformation $y^*x_* \Rightarrow p_*p^*$ of functors from $Spaces$ to $Spaces$. We note that both $y^*x_*$ and $p_*p^*$ are left exact functors and are hence corepresentable by pro-spaces, where the pro-space $Shp({\cal Z})$ corepresnting $p_*p^*$ is also known as the shape of ${\cal Z}$. Let $\hat{P}_{x,y} \in Pro(Spaces)$ denote the pro-space corepresenting $y^*x_*$. We then get that the data of a natural transformation $x_*p_* \Rightarrow y_*p_*$ is equivalent to the data of a map of pro-spaces $Shp({\cal Z}) \to \hat{P}_{x,y}$. We now recall that the formation of shapes ${\cal Z} \mapsto Shp({\cal Z})$ is left adjoint to the functor $\hat{X} \mapsto Spaces_{/\hat{X}}$ from pro-spaces to $\infty$-topoi. We may hence conclude that the data of a natural transformation $x_*p_* \Rightarrow y_*p_*$ is equivalent to the data of a geometric morphism ${\cal Z} \to Spaces_{/\hat{P}_{x,y}}$. We may then conclude that, if we let $q_*: Spaces_{/\hat{P}_{x,y}} \to Spaces$ be the terminal geometric morphism, then we have a canonical natural transformation $\tau:x_*q_* \Rightarrow y_*q_*$ which exhibits $Spaces_{/\hat{P}_{x,y}}$ as the desired comma object $Spaces \times^{\rm lax}_{\cal Y} Spaces$. Now let ${\cal P}_{x,y} \subseteq Spaces_{/\hat{P}_{x,y}}$ be the maximal left exact localization (see HTT 6.2.1.2) of $Spaces_{/\hat{P}_{x,y}}$ contained in the reflexive accessible subcategory $$\{X \in Spaces_{/\hat{P}_{x,y}} | \tau_X:x_*q_*X \to y_*q_*X \text{ is an equivalence}\} \subseteq Spaces_{/\hat{P}_{x,y}}.$$ Comparing universal properties we see that ${\cal P}_{x,y} \simeq Spaces \times_{\cal Y} Spaces$ represents the corresponding limit. In particular, for every points $x_*: Spaces \to {\cal Y}$ the loop $\infty$-topos ${\cal P}_{x,x} \simeq \Omega_x{\cal Y}$ is a left exact localization of a pro-étale $\infty$-topos.

Now suppose that ${\cal X}$ is an $\infty$-topos which is a double loop object, i.e., ${\cal X} \simeq \Omega_x{\cal Y}$ where ${\cal Y}$ itself is a loop object in $Topoi$. By the above we then have that ${\cal Y}$ is a left exact localization of pro-étale $\infty$-topos $Spaces_{/\hat{Y}}$, for some pro-space $\hat{Y} = \{Y_i\}_{i \in I} \in Pro(Spaces)$. Then $Spaces_{/\hat{Y}} = \lim_i Spaces_{/Y_i}$ and hence the space of points $y_*: Spaces \to Spaces_{/\hat{Y}}$ is naturally equivalent to the space $\lim_i Y_i = {\rm Map}_{Pro(Spaces)}(\ast,\hat{Y}) \in Spaces$. In this case, if $y_*: Spaces \to Spaces_{/\hat{Y}}$ corresponds to a compatible collection of points $y_i \in Y_i$ then
$$ \Omega_{y}Spaces_{/\hat{Y}} = \Omega_{y}\lim_i Spaces_{/Y_i} \simeq \lim_i \Omega_{y_i} Spaces_{/Y_i} \simeq \lim_i Spaces_{/\Omega_{y_i} Y_i} = Spaces_{/\Omega_{y}\hat{Y}} .$$ Furthermore, if $y_*: Spaces \to Spaces_{/\hat{Y}}$ is a point which factors as $Spaces \stackrel{x_*}{\to} {\cal Y} \hookrightarrow Spaces_{/\hat{Y}}$ then $\Omega_y (Spaces_{/\hat{Y}}) \simeq \Omega_x {\cal Y}$. It then follows that ${\cal X} \simeq \Omega_x {\cal Y} \simeq Spaces_{/\Omega_y\hat{Y}}$ is a pro-étale $\infty$-topos, as desired. $\Box$

Remarks:

1) At this point one may be tempted to conclude that every spectrum object in $Topoi$ is the image of a spectrum object in $Pro(Spaces)$. This is very possibly the case, but a-priori it does not follow from the above claim. All that can be deduced is that if we denote by $\widehat{Etale} \subseteq Topoi$ the full subcategory spanned by pro-étale topoi, i.e., the essential image of $Pro(Spaces) \to Topoi$, then every spectrum object in $Topoi$ comes from a spectrum object in $\widehat{Etale}$. However, since the functor $Pro(Spaces) \to \widehat{Etale}$ is not fully-faithful it is not a-priori clear if the map $Sp(Pro(Spaces)) \to Sp(\widehat{Etale})$ is essentially surjective. In other words, there could, in principle, be a spectrum object ${\cal X}_0,{\cal X}_1,...$ in $Topoi$ in which every ${\cal X}_i$ is a pro-étale spectrum ${\cal X}_1 \simeq Spaces_{/\hat{X}_i}$ but the structure equivalences $\varphi_i:{\cal X}_i \stackrel{\simeq}{\to} \Omega{\cal X}_{i+1}$ do not come from equivalences of pro-spaces $f_i:\hat{X}_i \stackrel{\simeq}{\to} \Omega\hat{X}_{i+1}$ (up to finitely many $\varphi_i$'s we can always arange it up to equivalence, but it's not clear if we can arrange all the $\varphi_i$'s at once; there is an obstruction to this which lies in a suitable $\lim^1$ set).

2) The claim that $Sh(S^1)$ is not an infinite loop object in $Topoi$ can be deduced from the above claim, at least if we assume that the truncation functor $\tau_{\leq 0}: Topoi \to Topoi_0$ from $\infty$-topoi to $0$-topoi (i.e., locales) preserves cofiltered limits (it seems to me that this claim should be deducible from the fact that cofiltered limits on both cases are computed in ${\rm Cat}_\infty$, see HTT 6.3.3.1). Assuming this, suppose that we had a CW complex $X$ such that $Sh(X)$ was an infinite loop object in $Topoi$. By the claim we have that $Sh(X) \simeq Spaces_{/\hat{X}} = \lim_i Spaces_{/X_i}$ for some pro-space $\hat{X} = \{X_i\}_{i \in I}$. By the commutativity of cofiltered limits and truncations we then have that the local $O(X)$ is equivalent to the local $\lim_i \tau_{\leq 0}(Spaces_{/X_i}) = \lim_i{\rm Sub}(\pi_0(X_i))$, where ${\rm Sub}(\pi_0(X_i))$ is the locale of subsets of $\pi_0(X_i)$. Since $X$ is Hausdorff it is sober and hence we may deduce that $X$ is homeomorphic to the limit $\lim_i \pi_0(X_i)$ (computed in topological spaces). But $X$ is a CW-complex and hence locally connected, and so $X \cong \lim_i \pi_0(X_i)$ would have to be a discrete set. In particular, $Sh(S^1)$ is not an infinite loop object (or even a double loop object).

As you already noticed there is a functor from the category of Spectrum to the category of topos-spectrum, whose image consist simply of étale topos.

As the category of toposes has all limits one easily see that the category of topos spectrum also has all limits (they are computed component wise).

The functor from spectrum to topos spectrum commutes to finite limits, so it can be extended into a limit preserving functor from the category of pro-spectrum to the category of topos-spectrum (the functor that compute the formal limits).

I believe that this functor produces plenty of non-étale example of topos spectrum.

For example, if I take a profinite abelian group $G$, which I see as a formal cofiltered limits of finite abelian group $G = lim_k G_k$ then each finite abelian group $G_k$ can be seen as a connective spectrum (you can put either $G_k$ in degree $0$ or $1$, it is not making a bit difference) and this produces an example of a pro-spectrum.

Its realiziation in topos-spectrum is the projective limits of the toposes $G_k$ (or $BG_k$ depending on the degree) and hence the limits is the category of sheaves on $G$ (or the $\infty$-topos completion of the ordinary topos $BG$ of smooth action of $G$)

This topos is not étale.

More generally, it is shown in Sketches of an elephant (C5.1.12) that the localic reflection functor (from $1$-topos to locales) preserve cofiltered limits. Assuming that this also applies to the $n$-localic reflection functor of Lurie this gives an easy way to compute the $n$-topos truncation of (the $k$-level of) the topos-spectrum attached to a prospectrum (as it will appears as a cofiltered limits of discrete $\infty$-topos). Computing $0$-topos or $1$-topos truncation of such topos might be an easy way to show that lots of example constructed this way are not étale.

---

Regarding $Sh(S^1)$ or other connected abelian topological group, they are going to be group-like $E_{\infty}$-algebra in the category of toposes but I don't think they can be delooped.

In $1$-topos theory, the loop space of a pointed topos, corresponds to the localic group of automorphisms of the point, which is always a totally disconnected localic group. So it can never be a connected group.

It is not completely clear what is the situation with $\infty$-toposes but I don't expect it will be very different. This might be related to this old question of mine and the answer by Jacob Lurie.

In fact because loop spaces of $\infty$-topos are expected to be totaly disconected and that there is a close connection between pro-abelian group and totally disconected abelian topological group, I suspect that there should be a close connection between topos-spectrum and pro-spectrum. (probably not an equivalence of categories, but not far from it). For example I can't think of any example of topos-spectrum that does not arise this way.