(2,1)-quasitopos?
structures in a cohesive (∞,1)-topos
For $(\Pi \dashv \Gamma \dashv LConst) : \mathbf{H} \to \infty Grpd$ a locally ∞-connected (∞,1)-topos and $X \in \mathbf{H}$ an object, we say that $\Pi(X)$ is the fundamental $\infty$-groupoid of $X$ in $\mathbf{H}$.
This is the object that encodes the geometric homotopy groups in an (∞,1)-topos.
For any $X \in \mathbf{H}$ the $\infty$-groupoid $\Pi(X)$ coincides with the fundamental ∞-groupoid of a locally ∞-connected (∞,1)-topos (see there) of the over-(∞,1)-topos $\mathbf{H}/X$.
In the cohesive (∞,1)-topos Top the intrinsic fundamental $\infty$-groupoid functor coincides with the ordinary fundamental ∞-groupoid of a topological space. See discrete ∞-groupoid for details.
In ETop∞Grpd the intrinsic fundamental $\infty$-groupoid is the generalization of that on Top to ∞-groupoids in paracompact spaces.
See also
for further discussion of the smooth shape modality of cohesion (the etale homotopy type operation in the context of smooth infinity-stacks) as applied to orbifolds and étale groupoids and generally étale ∞-groupoids.