(2,1)-quasitopos?
structures in a cohesive (∞,1)-topos
The analog in the context of (∞,1)-topos theory of a local geometric morphism in topos theory.
A local (∞,1)-geometric morphism $f : \mathbf{H} \to \mathbf{S}$ between (∞,1)-toposes $\mathbf{H},\mathbf{S}$ is
such that
a further right adjoint $f^! : \mathbf{S} \to \mathbf{H}$ to the direct image functor exists:
and $f$ is a ∞-connected (∞,1)-geometric morphism.
If $f : \mathbf{H} \to \mathbf{S}$ is the global section (∞,1)-geometric morphism in the over-(∞,1)-category Topos$/\mathbf{S}$, then we say that $\mathbf{H}$ is a local $(\infty,1)$-topos over $\mathbf{S}$.
If $\mathbf{S} =$ ∞Grpd then the extra condition that $f$ is ∞-connected (∞,1)-geometric morphism is automatic (see Properties – over ∞Grpd).
Every local $(\infty,1)$-topos over ∞Grpd has homotopy dimension $\leq 0$.
See homotopy dimension for details.
If an (∞,1)-geometric morphism $f : \mathbf{H} \to$ ∞Grpd has an extra right adjoint $f^!$ to its direct image, then $\mathbf{H}$ is an ∞-connected (∞,1)-topos.
By the general properties of adjoint (∞,1)-functors it is sufficient to show that $f_! f^* \simeq Id$. To see this, we use that every ∞-groupoid $S \in$ ∞Grpd is the (∞,1)-colimit (as discussed there) over itself of the (∞,1)-functor constant on the point: $S \simeq {\lim_\to}_{S} *$.
The left adjoint $f^*$ preserves all (∞,1)-colimits, but if $f_*$ has a right adjoint, then it does, too, so that for all $S$ we have
Now $f_*$, being a right adjoint preserves the terminal object and so does $f^*$ by definition of (∞,1)-geometric morphism. Therefore
Every local $(\infty,1)$-geometric morphism induces a notion of concrete (∞,1)-sheaves. See there for more (also see cohesive (∞,1)-topos).
Let $\mathbf{H}$ be any (∞,1)-topos (over ∞Grpd) and let $X \in \mathbf{H}$ be an object that is small-projective. Then the over-(∞,1)-topos $\mathbf{H}/X$ is local.
We check that the global section (∞,1)-geometric morphism $\Gamma : \mathbf{H}/X \to$ ∞Grpd preserves (∞,1)-colimits.
The functor $\Gamma$ is given by the hom-functor out of the terminal object of $\mathcal{H}/X$, this is $(X \stackrel{Id}{\to} X)$:
The hom-∞-groupoids in the over-(∞,1)-category are (as discussed there) homotopy fibers of the hom-sapces in $\mathbf{H}$: we have an (∞,1)-pullback diagram
Overserve that (∞,1)-colimits in the over-(∞,1)-category $\mathbf{H}/X$ are computed in $\mathbf{H}/X$.
If $X$ is small-projective then by definition we have
Inserting all this into the above $(\infty,1)$-pullback gives the $(\infty,1)$-pullback
By universal colimits in the (∞,1)-topos ∞Grpd, this (∞,1)-pullback of an (∞,1)-colimit is the $(\infty,1)$-colimit of the separate pullbacks, so that
So $\Gamma$ does commute with colimits if $X$ is small-projective. Since all (∞,1)-toposes are locally presentable (∞,1)-categories it follows by the adjoint (∞,1)-functor that $\Gamma$ has a right adjoint (∞,1)-functor.
local topos / local (∞,1)-topos.
and
The 1-categorical notion is discussed in