(2,1)-quasitopos?
structures in a cohesive (∞,1)-topos
If an (∞,1)-topos $\mathbf{H}$ is that of (∞,1)-sheaves on (the site of open subsets of) a paracompact topological space – $\mathbf{H} = Sh_{(\infty,1)}(X)$ – then its shape is the strong shape of $X$ in the sense of shape theory: a pro-object $Shape(X)$ in the category of CW-complexes.
It turns out that $Shape(X)$ may be extracted in a canonical fashion from just the (∞,1)-topos $Sh_{(\infty,1)}(X)$, and in a way that makes sense for any (∞,1)-topos. This then gives a definition of shape of general $(\infty,1)$-toposes.
The composite (∞,1)-functor
is the shape functor . Its value
on an $(\infty,1)$-topos $\mathbf{H}$ is the shape of $\mathbf{H}$.
Here
(∞,1)Topos is the (∞,1)-category of (∞,1)-toposes;
∞Grpd is the $(\infty,1)$-category of ∞-groupoids;
$Y$ is the (∞,1)-Yoneda embedding;
$Func(-,-)$ is the (∞,1)-category of (∞,1)-functors;
$AccLex(-,-) \subset (\infty,1)Func(-,-)$ is the full sub-(∞,1)-category of the (∞,1)-category of (∞,1)-functors on those which are left exact functors (preserve finite (∞,1)-limits) and also accessible.
$PSh(-) : \infty Grpd \to (\infty,1)Topos$ is the functor that produces the (∞,1)-category of (∞,1)-presheaves $Func(X^{op}, \infty Grpd)$ on $X$ (equivalently on the equivalent opposite ∞-groupoid $X^{op}$);
$Pro \infty Grpd$ is the (∞,1)-category of pro-objects in $\infty Grpd$.
That this does indeed land in accessible left exact functors is shown below.
Notice that for every (∞,1)-topos $\mathbf{H}$ there is a unique geometric morphism
where ∞Grpd is the $(\infty,1)$-topos of ∞-groupoids, $\Gamma$ is the global sections (∞,1)-functor and $LConst$ is the constant ∞-stack functor.
The shape of $\mathbf{H}$ is the composite functor
regarded as an object
For $X \in$ ∞Grpd we have by the (∞,1)-Grothendieck construction-theorem and using that up to equivalence every morphism of $\infty$-groupoids is a Cartesian fibration (see there) that
is the over-(∞,1)-category. Moreover, by the theorem about limits in ∞Grpd we have that the terminal geometric morphism $Hom(*,-): [X, \infty Grpd] \to \infty Grpd$ is the canonical projection $\infty Grpd/ X \to \infty Grpd$. This means that it is an etale geometric morphism. So for any geometric morphism $f : \mathbf{H} \to [X, \infty Grpd]$ we have a system of adjoint (∞,1)-functors
whose composite is the global section geometric morphism as indicated, because that is terminal.
Notice that in $\infty Grpd/X$ there is a canonical morphism
The image of this under $f^*$ is (using that this preserves the terminal object) a morphism
in $\mathbf{H}$.
Conversely, given a morphism of the form $* \to LConst X$ in $\mathbf{H}$ we obtain the base change geometric morphism
One checks that these constructions establish an equivalence
Using this, we see that
In particular this does show that $\Pi(\mathbf{H}) : \infty Grpd \to \infty Grpd$ does preserve finite $(\infty,1)$-limits, since $\Gamma$ preserves all limits and $LConst$ is a left exact functor. It also shows that it is accessible, since $\Gamma$ and $LConst$ are both accessible.
Suppose that $\mathbf{H}$ is locally ∞-connected, meaning that $\Gamma$ has a left adjoint $\Pi$ which constructs the homotopy ∞-groupoids of objects of $\mathbf{H}$. Then $\Shape(\mathbf{H})$ is represented by $\Pi(*)\in \infty Grpd$, for we have
Thus, if we regard $\Pi(*)$ as “the fundamental ∞-groupoid of $\mathbf{H}$” — which is reasonable since when $\mathbf{H}=Sh(X)$ consists of sheaves on a locally contractible topological space $X$, $\Pi_{\mathbf{H}}(*)$ is equivalent to the usual fundamental ∞-groupoid of $X$ — then we can regard the shape of an $(\infty,1)$-topos as a generalized version of the “homotopy $\infty$-groupoid” which nevertheless makes sense even for non-locally-contractible toposes, by taking values in the larger category of “pro-$\infty$-groupoids.”
It follows also that $\mathbf{H}$ is not only locally ∞-connected but also ∞-connected, then it has the shape of a point.
For a discussion of how the $(\infty,1)$-topos theoretic shape of $Sh_{(\infty,1)}(X)$ relates to the ordinary shape-theoretic strong shape of the topological space $X$ see shape theory.
The following is trivial to observe, but may be useful to note.
Let $(f_! \dashv f^* \dashv f_*) : \mathbf{H} \stackrel{\overset{f_!}{\to}}{\stackrel{\overset{f^*}{\leftarrow}}{\underset{f_*}{\to}}} \mathbf{B}$ be an essential geometric morphism of $(\infty,1)$-toposes that exhibits $\mathbf{B}$ as an essential retract of $\mathbf{H}$ in that
Then the shape of $\mathbf{B}$ is equivalent to that of $\mathbf{H}$.
Since $\infty Grpd$ is the terminal object in the category of Grothendieck $(\infty,1)$-toposes and geometric morphisms, we have
Every
and hence also every cohesive (∞,1)-topos
over $\infty Grpd$ has the shape of the point.
By definition $\mathbf{H}$ is $\infty$-connected if the constant ∞-stack inverse image $f^* = L Const$ is
not only a left but also a right adjoint;
By standard properties of adjoint (∞,1)-functors we have that a right adjoint $f^*$ is a full and faithful (∞,1)-functor precisely if the counit $f_! f^* \to Id$ is an equivalence.
Equivalently, we can observe that a locally ∞-connected (∞,1)-topos is ∞-connected precisely when $\Pi$ preserves the terminal object, and apply the above observation that the shape of a locally ∞-connected (∞,1)-topos is represented by $\Pi(*)$.
shape of an $(\infty,1)$-topos
The definition of shape of $(\infty,1)$-toposes as $\Gamma \circ LConst$ is due to
This and the relation to shape theory, more precisely the strong shape, of topological spaces is further discussed in section 7.1.6 of