(2,1)-quasitopos?
structures in a cohesive (∞,1)-topos
An (∞,1)-topos $\mathcal{X}$ has homotopy dimension $\leq n \in \mathbb{N}$ if every (n-1)-connected object $A$ has a global element, a morphism $* \to A$ from the terminal object into it.
This appears as HTT, def. 7.2.1.1.
An (∞,1)-topos $\mathcal{X}$ is locally of homotopy dimension $\leq n \in \mathbb{N}$ if there exists a collection $\{U_i \in \mathcal{X}\}$ of objects such that
the $\{U_i\}$ generate $\mathcal{X}$ under (∞,1)-colimits;
each over-(∞,1)-topos $\mathcal{X}/U_i$ has homotopy dimension $\leq n$.
This appears as HTT, def. 7.2.1.8.
If an (∞,1)-topos $\mathcal{X}$ is locally of homotopy dimension $\leq n$ for some $n \in \mathbb{N}$ then it is a hypercomplete (∞,1)-topos.
This appears as HTT, cor. 7.2.1.12.
If $\mathcal{X}$ has homotopy dimension $\leq n$ then it also has cohomological dimension $\leq n$.
The converse holds if $\mathcal{X}$ has finite homotopy dimension and $n \geq 2$.
This appears as HTT, cor. 7.2.2.30.
An (∞,1)-topos $\mathcal{X}$ has homotopy dimension $\leq n$ precisely if the global section (∞,1)-geometric morphism
has the property that $\Gamma$ sends $(k\geq n)$-connective morphisms to $(k-n)$-connective morphisms.
This is HTT, lemma 7.2.1.7
Up to equivalence, the unique (∞,1)-topos of homotopy dimension $\leq -1$ is the the terminal category $* \simeq Sh_{(\infty,1)}(\emptyset)$.
This is HTT, example. 7.2.1.2.
An object $X \in \mathcal{X}$ is $(-1)$-connected if the morphism $X \to *$to the terminal object in an (∞,1)-category is. This is the case if it is an effective epimorphism.
Since the global section (∞,1)-functor is corepresented by the terminal object, $X$ is 0-connective precisely if $\Gamma(X) \to \Gamma(*) = *$ is an epimorphism on connected components. By the discussion at effective epimorphism, this is the case precisely if $\Gamma(X) \to *$ is an effective epimorphism in ∞Grpd.
So $\mathcal{X}$ has homotopy dimension $\leq 0$ if $\Gamma$ preserves effective epimorphisms. This is the case if it preserves finit (∞,1)-limits (the (∞,1)-pullbacks defining a Cech nerve) and all (∞,1)-colimits (over the resulting Cech nerve). being a right adjoint (∞,1)-functor $\Gamma$ always preserves (∞,1)-limits. If $\mathcal{X}$ is local then $\Gamma$ is by definition also a left adjoint and hence also preserves (∞,1)-colimits.
Every local (∞,1)-topos has homotopy dimension $\leq 0$.
Let
be the terminal geometric morphism of the local $(\infty,1)$-topos, with $\nabla$ being the extra right adjoint to the global section (∞,1)-geometric morphism functor that characterizes locality.
By prop it is sufficient to show that $\Gamma$ send (-1)-connected morphisms to (-1)-connected morphisms, hence effective epimorphisms to effective epimorphisms.
By the existence of $\nabla$ we have that $\Gamma$ preserves not only (∞,1)-limits but also (∞,1)-colimits. Since effective epimorphisms are defined as certain colimits over diagrams of certain limits, $\Gamma$ preserves effective epimorphisms.
So in particular for $C$ any (∞,1)-category with a terminal object, the (∞,1)-category of (∞,1)-presheaves $PSh_{(\infty,1)}(C)$ is an (∞,1)-topos of homotopy dimension $\leq 0$. Notably Top $\simeq$ ∞Grpd $\simeq PSh_{(\infty,1)}(*)$ has homotopy dimension $\leq 0$.
This is HTT, example. 7.2.1.3.
Every (∞,1)-category of (∞,1)-presheaves is an (∞,1)-topos of local homotopy dimension $\leq 0$.
This appears as HTT, example. 7.2.1.9.
If a paracompact topological space $X$ has covering dimension $\leq n$, then the (∞,1)-category of (∞,1)-sheaves $Sh_{(\infty,1)}(X) := Sh_{(\infty,11)}(Op(X))$ is an (∞,1)-topos of homotopy dimension $\leq n$.
This is HTT, theorem 7.2.3.6.
For $X \in$ ∞Grpd $\simeq$ Top an object, the over-(∞,1)-topos $\infty Grpd/X$ has homotopy dimension $\leq n$ precisely if $X \in Top$ a retract in the homotopy category $Ho(Top)$ of a CW-complex of dimension $\leq n$.
This is HTT, example 7.2.1.4.
notion of dimension
The (∞,1)-topos theoretic notion is discuss in section 7.2.1 of