For any spectrum $E$, there is a "discrete" topos spectrum $(Spaces / E\_n)\_n$. And I believe any topos spectrum is a localization of a "discrete" one. Are there any "non-discrete" topos spectra?


To be precise, let $Topoi$ be the $\infty$-category of $\infty$-topoi and geometric morphisms (pointing in the direction of the right adjoint). Note that $Spaces$ is terminal in $Topoi$. Let $Sp(Topoi)$ be the stabilization of this category: an object is a sequence $(E\_n)\_{n \in \mathbb Z}$ of pointed $\infty$-toposes equipped with equivalences $E\_n = \Omega E\_{n+1}$ where $\Omega$ is the loops functor on the category $Topoi\_\ast$ of pointed toposes. The question is


**Question:** Are there examples of objects of $Sp(Topoi)$ other than the "discrete" ones referred to above?


As alluded to above, I think there is a major restriction on the possibilities: the $\Omega$ functor on pointed $\infty$-categories lands in spaces. Because the "presheaves" functor preserves limits, the $\Omega$ functor on presheaf toposes lands in slices of $Spaces$. Since every $\infty$-topos embeds in a presheaf topos and embeddings are stable under finite limits, this implies that if $(E\_n)\_n$ is a topos spectrum, then each $E\_n$ embeds in (i.e. is a localization of) a slice of $Spaces$. I think this sounds pretty restrictive -- I don't know an example of a topos that embeds into a slice of $Spaces$ which is not itself a slice of $Spaces$!



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). 


