(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 $\LConst$ 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 the following sequence of natural equivalences of ∞-groupoids:
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
Program Stacks, Intersection theory and Non-abelian Hodge Theory , MSRI, Berkeley, January-May 2002 (arXiv:math/0212330).
This and the relation to shape theory, more precisely the strong shape, of topological spaces is further discussed in section 7.1.6 of
See also
Last revised on December 6, 2019 at 09:43:36. See the history of this page for a list of all contributions to it.