(2,1)-quasitopos?
structures in a cohesive (∞,1)-topos
(see also Chern-Weil theory, parameterized homotopy theory)
For $\mathbf{H}$ an (∞,1)-topos and $X \in \mathbf{H}$ any object, the over-(∞,1)-category $\mathbf{H}/X$ is itself an $(\infty,1)$-topos: the over-$(\infty,1)$-topos .
If we think of $\mathbf{H}$ as a big topos, then for $X \in \mathbf{H}$ we may think of $\mathbf{H}/X \in$ (∞,1)-topos as the little topos-incarnation of $X$. The objects of $\mathbf{H}/X$ we may think of as (∞,1)-sheaves on $X$.
This correspondence between objects of $X$ and their little-topos incarnation is entirly natural: $\mathbf{H}$ is equivalently recovered as the (∞,1)-category whose objects are over-$(\infty,1)$-toposes $\mathbf{H}/X$ and whose morphisms are (∞,1)-geometric morphisms over $\mathbf{H}$.
For $\mathbf{H}$ an (∞,1)-topos and $X \in \mathbf{H}$ an object also the over-(∞,1)-category $\mathbf{H}/X$ is an $(\infty,1)$-topos. This is the over-$(\infty,1)$-topos of $\mathbf{H}$ over $X$.
This is HTT, prop 6.3.5.1 1).
There is a canonical (∞,1)-geometric morphism
where the extra left adjoint $X_!$ is the obvious projection $X_! : (Y \to X) \mapsto X$, and $X_*$ is given by forming the (∞,1)-product with $X$.
This is called an etale geometric morphism. See there for more details.
The fact that $(X_! \dashv X^*)$ follows from the universal property of the products. The fact that $X^*$ preserves (∞,1)-colimits and hence has a further right adjoint $X_*$ by the adjoint (∞,1)-functor theorem follows from that fact that $\mathbf{H}$ has universal colimits.
If $\mathbf{H}$ is a locally ∞-connected (∞,1)-topos then for all $X \in \mathbf{H}$ also the over-$(\infty,1)$-topos $\mathbf{H}/X$ is locally $\infty$-connected.
The composite of (∞,1)-geometric morphisms
is itself a geometric morphism. Since ∞Grpd is the terminal object in (∞,1)Topos this must be the global section geometric morphism for $\mathbf{H}/X$. Since it has the extra left adjoint $\Pi \circ X_!$ it is locally $\infty$-connected.
Let $((\infty,1)Topos/\mathbf{H})_{et} \subset (\infty,1)Topos/\mathbf{H}$ be the full sub-(∞,1)-category on the etale geometric morphisms $\mathbf{H}/X \to \mathbf{H}$. Then there is an equivalence
See etale geometric morphism for more details.
See base change geometric morphism.
We spell out how $\mathbf{H}/X$ is the (∞,1)-category of (∞,1)-sheaves over the big site of $X$.
(forming overcategories commutes with passing to presheaves)
Let $C$ be a small (∞,1)-category and $X : K \to C$ a diagram. Write $C_{/X}$ and $PSh(C)/_{X}$ for the corresponding over (∞,1)-categories, where – notationally implicitly – we use the (∞,1)-Yoneda embedding $C \to PSh(C)$.
Then we have an equivalence of (∞,1)-categories
This appears as HTT, 5.1.6.12. For more on this see (∞,1)-category of (∞,1)-presheaves.
Here we may think of $C/X$ as the big site of the object $c \in PSh(C)$, hence of $PSh(C/X)$ as presheaves on $X$.
Let $C$ be equipped with a subcanonical coverage, let $X \in C$ and regard $C/X$ as an (∞,1)-site with the big site-coverage. Then we have
By the discussion of adjoint (∞,1)-functors on over-(∞,1)-categories adjoint (∞,1)-functors we have that the adjunction
passes to an adjunction on the over-(∞,1)-categories
(where we are using that $F i X \simeq X$ by the assumption that the coverage is subcanonical, so that the representable $X$ is a (∞,1)-sheaf), such that $i/X$ is still a full and faithful (∞,1)-functor (where we are using that the unit $X \to F i X$ is an equivalence, since $X$ is a sheaf).
Since moreover the (∞,1)-limits in $Sh(C)/X$ are computed as limits in $Sh(C)$ over diagrams with a bottom element adjoined (as discussed at limits in over-(∞,1)-categories) it follows that with $F$ preserving all finite limits, so does $F/X$.
In summary we have that $(F/X \dashv i/X)$ is a reflective sub-(∞,1)-category of $PSh(C/X)$ hence is the (∞,1)-category of (∞,1)-sheaves on the category $C/X$ for some (∞,1)-site-structure. But since $F/X$ inverts precisely those morphisms that are inverted by $F$ under the projection $PSh(C)/X \to PSh(C)$, it follows that this is the big site structure on $C/X$ (this is the defining property of the big site).
Specifically for the $(\infty,1)$-topos $\mathbf{H} =$ ∞Grpd we also have the following characterization.
For $\mathbf{H} =$ ∞Grpd we have that for $X \in \infty Grp$ any ∞-groupoid the corresponding over-$(\infty,1)$-topos is equivalent to the (∞,1)-category of (∞,1)-presheaves on $X$:
This is a special case of the (∞,1)-Grothendieck construction. See the section (∞,0)-fibrations over ∞-groupoids.
The following proposotion asserts that the over-$(\infty,1)$-topos over an $n$-truncated object indeed behaves like a generalized n-groupoid
For $n \in \mathbb{N}$ and $\mathcal{X}$ an n-localic (∞,1)-topos, then the over-$(\infty,1)$-topos $\mathcal{X}/U$ is $n$-localic precisely if the object $U$ is $n$-truncated.
This is (StrSp, lemma 2.3.16).
If $Obj_\kappa \in \mathbf{H}$ is an object classifier for $\kappa$-small objects, then the projection $Obj_\Kappa \times X \o X$ regarded as an object in the slice is a $\kappa$-small object classifier in $\mathbf{H}_{/X}$.
If a homotopy type theory is the internal language of $\mathbf{H}$, then then theory in context $x : X \vdash \cdots$ is the internal language of $\mathbf{H}_{/X}$.
over-(∞,1)-topos
The general notion is discussed in section 6.3.5 of
Some related remarks are in