Let $T$ be a topos over a site, such as the étale topos $\operatorname{Et}_{S}$ over a scheme $S$. This topos comes with a cohomology theory $H^{\bullet}$ and the notion of homotopy groups $\pi_{\bullet}$.
Assume I have an object $X\in S$, and I want to calculate its cohomology. If all higher homotopy groups $\pi_{n}(X)$, $n\geq 2$, vanish, this cohomology is the same as the cohomology of the fundamental group: \begin{equation*} H^{i}(X,F)=H^{i}(\pi_{1}(X),F) \end{equation*} and one says that $X$ is a classifying object of its fundamental group.
This is for instance the case when $T=\operatorname{Et}_{S}$ and $X=\operatorname{Spec}(k)$ for some field $k$, see i.e. Fu, Étale Cohomology, Thm. 5.7.8.
This gives rise to the following questions:
(1.) To what degree does the converse hold? Are the spectra of fields (and direct products thereof) the only classifying objects in the category $Ét_{S}$?
(2.) Is there a way to describe which objects in a given topos $T$ are classifying?
(3.) If $C$ is a CW-complex, we can turn $C$ into a classifying space by adding cells in such a way that the higher homotopy groups get killed. If $C$ is the classifying space of a topos $T$, this amounts to adding objects and morphisms. There exists therefore a "smallest" topos $\tilde{T}$, well-defined up to homotopy, containing $T$ so that $X$ is classifying in $\tilde{T}$. Is there an intrinsic way to describe $\tilde{T}$?
