(2,1)-quasitopos?
structures in a cohesive (∞,1)-topos
Every (∞,1)-topos $E$ has a shape $Shape(E) \in Pro\infty Grpd$. When $E$ is locally ∞-connected then this is a genuine ∞-groupoid $\Pi(E) \in$ ∞Grpd. We may think of this as the fundamental ∞-groupoid of the $(\infty,1)$-topos regarded as a generalized space.
But also every locally ∞-connected (∞,1)-topos has an internal notion of fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos for objects of $E$, denoted $\Pi_E : E \to \infty Grpd$. Applied to its terminal object this does agree with the fundamental ∞-groupoid of the topos:
Conversely, for an object $X\in E$, the fundamental ∞-groupoid $\Pi_E(X)$ internal to $E$ can be identified with the fundamental ∞-groupoid of the locally ∞-connected (∞,1)-topos $E/X$.
For $(\Pi_E \dashv \Gamma_E \dashv LConst_E) : E \to \infty Grpd$ a locally ∞-connected (∞,1)-topos we say its fundamental $\infty$-groupoid is
where $*$ is the terminal object of $E$.
In other words, it is the internal fundamental ∞-groupoid of the terminal object of $E$.
Let $\mathbf{H}$ be a locally $\infty$-connected $(\infty,1)$-topos and $X \in \mathbf{H}$ an object. Then also the over-(∞,1)-topos $\mathbf{H}/X$ is locally $\infty$-connected (as discussed there).
We have then two different definitions of the fundamental $\infty$-groupoid of $X$: once as $\Pi_{\mathbf{H}}(X)$ – the fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos – and once as $\Pi(\mathbf{H}/X)$.
These agree:
Since $X \stackrel{Id}{\to} X$ is the terminal object in $\mathbf{H}/X$ we have by definition
Now observe that $\Pi_{\mathbf{H}/X} = \Pi_{\mathbf{H}} \circ X_!$ since the terminal global section geometric morphism of the over-topos is
and that $X_!$ in the etale geometric morphism is the projection map that sends $Y \stackrel{}{\to} X$ to $Y$.
Let $LC(\infty,1)Topos$ denote the full sub-(∞,1)-category of (∞,1)Topos determined by the locally ∞-connected objects.
The $(\infty,1)$-category $\infty Gpd$ (as the category of local homeomorphisms over $\infty Gpd$) is reflective in $LC(\infty,1)Topos$,
with the reflector given by forming the fundamental $\infty$-groupoid.
Any ∞-groupoid $G$ gives rise to an (∞,1)-presheaf (∞,1)-topos $PSh(G) = [G^{op},\infty Gpd]$, which by the (∞,1)-Grothendieck construction is equivalent to the over-(∞,1)-topos $\infty Gpd / G$. The $(\infty,1)$-toposes of this form are, by definition, those for which the unique (∞,1)-geometric morphism to $\infty Gpd$ is a local homeomorphism of toposes. This construction embeds $\infty Gpd$ as a full sub-(∞,1)-category of (∞,1)Topos:
since in particular the $(\infty,1)$-toposes $PSh(G)$ are locally ∞-connected.
To show that $\Pi$ is a left adjoint (∞,1)-functor to $PSh(-)$ we demonstrate a natural hom-equivalence
for $E\in LC(\infty,1)Topos$ and $A \in \infty Grpd$.
At shape of an (∞,1)-topos it is shown that we have a natural equivalence
Now observe that furthermore we have a sequence of natural equivalences
So equivalently, one may say that a locally ∞-connected (∞,1)-topos $E$ has a shape which is representable, and its fundamental ∞-groupoid $\Pi(\mathbf{H})$ is the representing object.
For $X$ a locally contractible topological space, we have an equivalence
between the ordinary fundamental ∞-groupoid of $X$ defined by the singular simplicial complex and the topos-theoretic fundamental $\infty$-groupoid of the (∞,1)-sheaf (∞,1)-topos $(\infty,1)Sh(X)$ over $X$.
Details are at geometric homotopy groups in an (∞,1)-topos.
More generally the shape of an (∞,1)-topos of $(\infty,1)Sh(X)$ reproduces the shape theory of $X$.
fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos / of a locally $\infty$-connected $(\infty,1)$-topos