(2,1)-quasitopos?
structures in a cohesive (∞,1)-topos
An (∞,1)-topos is hypercomplete if the Whitehead theorem is valid in it.
Equivalently: if all its object are hypercomplete objects.
Every (∞,1)-topos has a hypercompletion, given by the full reflective sub-(∞,1)-category spanned by its hypercomplete objects.
For $\mathcal{X}$ an (∞,1)-topos, a point of $\mathcal{X}$ is a geometric morphism
from ∞Grpd $\simeq Sh_\infty(\ast)$ to $\mathcal{X}$. We say that $\mathcal{X}$ has enough points if a 1-morphism $f \colon X \to Y$ in $\mathcal{X}$ in an equivalence in $\mathcal{X}$ precisely if for all such points the inverse image $p^\ast (f)$ (the stalk at the point) is an equivalence in ∞Grpd.
There exist 1-sites $C$ such that the (1,1)-topos of sheaves of sets on $C$ has enough points in the 1-topos sense, but such that the corresponding 1-localic (∞,1)-topos $\mathcal{X}$ does not have enough points in the sense of def. 2. An example is given by the site of open subsets of the topological space $\prod_{\mathbb{N}} \{x,z,y\}$ where the topology on $\{x,z,y\}$ is generated by the two open subsets $\{x,y\}$ and $\{x,z\}$. See HTT, Remark 6.5.4.7.
However, the hypercompletion $\mathcal{X}^\wedge$ of $\mathcal{X}$ will have enough points. This follows from the fact that $\mathcal{X}^\wedge$ may be presented by the Jardine model structure on simplicial presheaves on the given site of definition, and that in the presence of enough 1-topos points the weak equivalences of that model structure are the stalk-wise weak equivalences in the model structure on simplicial sets. (See at model structure on simplicial presheaves for details.)
An $(\infty,1)$-topos that has enough points is hypercomplete.
This is HTT, remark 6.5.4.7.
Recall from def. 2 that a point of an (∞,1)-topos $\mathbf{H}$ is a geometric morphism $p : \infty Grpd \stackrel{\overset{p^*}{\leftarrow}}{\underset{p_*}{\to}} \mathbf{H}$.
And by definition $\mathbf{H}$ has enough points if a morphism $F : X \to Y$ in $\mathbf{H}$ is an equivalence if for all points $p$, the stalk $p^* f$ is an equivalence in ∞ Grpd.
But if $f$ is $\infty$-connected in $\mathbf{H}$, then so is $p^* f$ in ∞Grpd, which is hypercomplete, by Whitehead's theorem, so that $p^* f$ is an equivalence.
So in an $(\infty,1)$-topos with enough points all $\infty$-connected morphisms are equivalences.
All geometric morphisms $X \to Y$ out of a hypercomplete $(\infty,1)$-topos $X$ factor through the hypercompletion $\hat Y$ of $Y$:
the inclusion $\hat Y \hookrightarrow Y$ induces an equivalence
This is HTT, prop. 6.5.2.13.
Every (∞,1)-topos which is locally of homotopy dimension $\leq n$ for some finite $n\geq -1$ is hypercomplete.
See the discussion at homotopy dimension for details and further implications.
Hypercomplete ∞-stack (∞,1)-toposes are precisely those that are presented by the local Joyal-Jardine model structure on simplicial presheaves, where weak equivalences of simplicial presheaves are those morphisms that induce isomorphisms on homotopy sheaves. In these models the fibrant objects are those simplical presheaves that satisfy descent over all hypercovers.
If the underlying ordinary sheaf topos has enough points, then this are equivalently the morphisms that induce stalkwise weak equivalences in the standard model structure on simplicial sets.
Contrary to that one can consider local models by left Bousfield localization of the global model structure on simplicial presheaves only at Cech covers. This yield in general a non-hypercomplete (∞,1)-categories of (∞,1)-sheaves.
For $C$ site, write $[C^{op}, sSet]_{inj,loc}$ for the Joyal-Jardine local model structure on simplicial presheaves (whose weak equivalence are morphisms that induce isomorphisms on homotopy-sheaves).
The $(\infty,1)$-topos that this presents is the hypercompletion of the (∞,1)-category of (∞,1)-sheaves on $C$:
This is HTT, prop. 6.5.2.14.
The strategy is to form the localization in a 2-step process, where we first just form the Cech-localization, and then from that the full hypercompletion. For that notice that among the weak equivalences in the Joyal-Jardine local model structure on simplicial presheaves are in particular the ordinary covering sieves $S(\{U_i\}) \hookrightarrow j(U)$ (here $j$ is the Yoneda embedding) associated with a coverin family $\{U_i \to U\}$ in the site $C$:
since $C$ is an ordinary category, the simplicial presheaves $S(\{U_i\})$ and $j(U)$ have vanishing presheaves of homotopy groups in positive degree, while they coindide with their $\pi_0$-presheaves. Since the sheafification of $S(\{U_i\})$ is isomorphic to $j(U)$, by definition, it follows that the same holds for the $\pi_0$-presheaves and trivially for the $\pi_n$-presheaves. So $S(\{U_i\}) \to j(U)$ is a Joyal-Jardine weak equivalence.
We will now first localize with respect to these morphisms to obtain the Cech-localization whose fibrant objects are (∞,1)-sheaves. The point is that on these fibrant objects then, the Joyal-Jardine sheaves of homotopy groups can be seen to coincide with the (∞,1)-categorical homotopy sheaves in terms of which hypercompletion is defined.
To start with, as discussed at (∞,1)-category of (∞,1)-presheaves we have that the global model structure presents the $(\infty,1)$-presheaves:
Observe that
every simplicially constant object is fibrant in $[C^{op}, sSet]_{inj}$ ;
hence since every object is cofibrant, the morphism $S(\{U_i\}) \to j(U)$ is in $([C^{op}, sSet]_{inj})^\circ$;
under the abovee identification it is an (∞,1)-monomorphism in $PSh_{(\infty,1)}(C)$
(discuss this bit in more detail…).
Therefore the topological localization of $PSh_{(\infty,1)}(C)$ at these monomorphisms, i.e. the (∞,1)-category of (∞,1)-sheaves on $C$ is presented by the left Bousfield localization $[C^{op}, sSet]_{inj,cov}$ of $[C^{op}, sSet]_{inj}$ at the covering subfunctors $S(\{U_i\}) \to j(U)$.
By the above remark, the Joyal-Jardine localization $[C^{op}, sSet]_{proj,loc}$ that we are after is a further localization of this Cech localization : we have the bottom row in the following diagram, and want to see that the top left corner is as indicated:
Now recall that the categorical homotopy groups in an (∞,1)-topos of an object $X$ are defined by first forming the powering $X^{* \to S^n} : X^{S^n} \to X$ and then paassing to the 0-truncation $\tau_{\leq 0} ( X^{S^n} \to X)$ of this object in the over (∞,1)-category.
By the discussion at Tensoring and cotensoring with an ∞-groupoid we have that this powering operation is on fibrant objects modeled by the powering in the sSet-enriched model category $[C^{op}, sSet]_{inj,cov}$. But the powering of simplicial presheaves by simplicial set is just objectwise the internal hom of simplicial sets. In terms of this are defined the objectwise simplicial homotopy groups and hence the Joyal-Jardine homotopy-presheaves.
Furthermore, if $X \in [C^{op}, sSet]_{inj,cov}$ is fibrant, it satisfies descent for simplicial presheaves at Cech covers. Since powering is a Quillen bifunctor, the same is then true for $X^S$, formed in the model category, so $X^S$ is an $\infty$-stack. But that means its 0-truncation $\tau_{\leq 0}(X^{S^n})$ is an ordinary sheaf. (Observe that truncation commutes with localization, as discussed here.)
In total this shows that on fibrant objects $X$ in $[C^{op}, sSet]_{inj,cov}$, the Joyal-Jardine homotopy sheaves coincide with the $(\infty,1)$-categorical homotopy sheaves of the object $X$.
It remains to observe that under left Bousfield lcoalization, the new fibrant objects are precisely those old fibrant objects that are also local objects with respect to the morphisms at which one localizes. With the above this implies that the left Bousfield localization $[C^{op}, sSet]_{inj,cov} \to [C^{op}, sSet]_{inj,loc}$ does model the hypercompletion $Sh_{(\infty,1)}(C) \to \widehat {Sh_{(\infty,1)}}(C)$.
In classical topos theory literature frequently simplicial objects in an ordinary topos are considered, with acyclic fibrations taken to be those morphisms $Y_\bullet \to X_\bullet$ such that for all horn inclusions the induced morphism
is an epimorphism in the topos.
See for instance page 17 of
(and it looks like this is the discussion planned for part E of the Elephant).
For sheaf toposes epimorphism means stalk-wise epimorphism. Therefore this amounts to using on simplicial sheaves the structure of a category of fibrant objects as defined in BrownAHT, where acyclic fibrations are the stalkwise acyclic Kan fibrations.
The homotopy category of this homotopical category is the same as that of the Joyal-Jardine model structure on simplicial presheaves in the presence of enough points (since in both cases weak equivalences are the stalkwise weak equivalences), hence is the same as the homotopy category of the hypercomplete (∞,1)-topos.
For more discussion of how this classical definition interplays with other definitions see also homotopy groups in an (∞,1)-topos.
The notion of hypercompleteness appears as $t$-completeness in
The notion of hypercomplete $(\infty,1)$-toposes is the topic of section 6.5 of