As discussed in the comments, I'm writing here the proof of the following fact:

Let $\mathscr{X}_∞$ be the ∞-topos of sheaves on $\mathrm{FinTop}^{op}$ under the atomic topology (the topology where a sieve is covering iff it is nonempty). Then for any other ∞-topos $\mathscr{X}$, there is a natural equivalence between the ∞-category of geometric morphisms $\mathscr{X}\to \mathscr{X}_∞$ and the ∞-category of ∞-connective objects in $\mathscr{X}$.

To do so, we will need a lemma.

**Lemma:** Let $\mathscr{X}$ be an ∞-topos and let $X\in\mathscr{X}$ be an object. Then all the following statements are equivalent

- The object $X$ is ∞-connected. That is $\pi_nX\to X$ is an equivalence for each $n\ge 0$ and $X\to *$ is an effective epimorphism.
- For every $n\ge -1$ the diagonal map $X\to X^{S^n}$ is an effective epimorphism (where $S^{-1}=\varnothing$)
- For every finite space $T$ the diagonal map $X\to X^T$ is an effective epimorphism.
- For every nonempty collection of maps of finite spaces $\{T\to T'_i\}_{i\in I}$ the map $\coprod_{i\in I} X^{T_i'}\to X^T$ is an effective epimorphism.

The lemma immediately implies the fact we were looking for thanks to proposition 6.2.3.20 in Higher Topos Theory. In fact our condition 4 is exactly the condition that the left exact functor $\mathrm{FinTop}^{op}\to \mathscr{X}$
$$T\mapsto X^T$$
sends covering families to effective epimorphisms.

*Proof:* $4 \Rightarrow 3\Rightarrow 2$ is obvious. Let us prove $1\Leftrightarrow 2$. We know that $\pi_nX$ is the 0-truncation of $X^{S^n}\to X$ given by the evaluation at the basepoint. So $\pi_nX\to X$ is an equivalence iff $X^{S^n}\to X$ is a 0-connected. But by HTT.6.5.1.20, this is true iff the diagonal map $X\to X^{S^n}$ (which is a section of the evaluation) is -1-connected, i.e an effective epimorphism.

In the following we will often use HTT.6.2.3.12, that is if $gf$ is an effective epimorphism, so is $g$.

$2\Rightarrow 3$ This follows from an induction on the number of cells of $T$. Let us assume that it is true for $T$ and we will show it is true for $T'=T\amalg_{S^{n-1}}D^n$. There is a cofiber sequence
$$T\to T'\to T'/T=S^n$$
and so we obtain a pullback square

$$\require{AMScd}
\begin{CD}
X^{S^n} @>>> X^{T'}\\
@VVV @VVV\\
X @>>> X^T
\end{CD}
$$
Since effective epimorphisms are stable under pullbacks (HTT.6.2.3.15), it follows that $X^{S^n}\to X^{T'}$ is an effective epimorphism. Finally, since $X\to X^{S^n}$ is an effective epimorphism by hypothesis and by HTT.6.2.3.12 effective epimorphisms are stable under composition, we are concluded.

$3 \Rightarrow 4$ This follows from HTT.6.2.3.12 applied to the compositions
$$ X\to X^{T_j'}\to \amalg_{i\in I}X^{T_i'}\to X^T$$
for some $j\in I$. $\square$

doesthe $\infty$-topos of local systems of spectra classify? $\endgroup$pointedobjects with some property. So it can't be exactly the classifying topos you're looking for. $\endgroup$5more comments