(2,1)-quasitopos?
structures in a cohesive (∞,1)-topos
To the extent that an (∞,1)-topos may be thought of as representing a space, the shape of the $\infty$-topos is the underlying pro-homotopy type of this space (Def. below). Equivalently, the shape of an $\infty$-topos is the generalized étale homotopy-type of its terminal object (Def. , Prop. below).
In the special case that the (∞,1)-topos is that of (∞,1)-sheaves on (the site of open subsets of) a paracompact topological space, its shape coincides with the strong shape of $X$ in the classical sense of shape theory.
In the special case that the $\infty$-topos is a slice of a cohesive $(\infty,1)$-topos $\mathbf{H}$ over some object, its shape is the cohesive shape of that object (Prop. below).
We state three somewhat different-looking definitions, and show that they are all equivalent to each other.
(shape of an $\infty$-topos – Toën-Vezzosi 2002, Def. 5.3.2)
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 follows from the equivalence to the following definition, see Rem. below.
Notice (see here) 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
$LConst$ is the constant ∞-stack functor.
(shape of an $\infty$-topos – Lurie 2009, Def. 7.1.6.3)
The shape of an $(\infty,1)$-topos $\mathbf{H}$ is the composite $(\infty,1)$-functor
regarded as a pro-$\infty$-groupoid:
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 slice (∞,1)-category. Moreover, by this 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 $\mathbf{H}$ of the form
Conversely, given a morphism of the form $* \to LConst X$ in $\mathbf{H}$ we obtain the base change geometric morphism of slice $(\infty,1)$-toposes
One checks that these constructions establish an equivalence
Using this, we find the following sequence of equiavlences:
The composite of these is the equivalence to be shown.
Prop. immediately implies that $Shp(\mathbf{H}) \;\colon\; \infty Grpd \to \infty Grpd$ according to Def. does preserve finite $(\infty,1)$-limits, since in the equivalent Def. this is manifest: There $\Gamma$ clearly preserves all limits, since it is a right adjoint, and $LConst$ preserves finite limits, since it is a left exact functor by definition. Similarly, this makes manifest that $Shp(\mathbf{H})$ is accessible, since $\Gamma$ and $LConst$ are both accessible.
(shape of an $\infty$-topos – Hoyois 2013, p. 3)
For $(\Gamma \dashv LConst) \;\colon\; \mathbf{H} \to \infty Grpd$ an $(\infty,1)$-topos, write (with convenient overloading of notation)
for the pro-left adjoint to $LConst \,\colon\, \infty Grpd \to \mathbf{H},$ and say that the shape of $\mathbf{H}$ is the image under this pro-left adjoint of its terminal object $\ast_{\mathbf{H}}$:
Consider the following sequence of natural equivalences:
Here the line is just Def. (alternatively: is Prop. ), and the second line follows by the cartesian closure of ∞Grpd. The third line is the characteristic hom-equivalences of the adjunction $\Gamma \dashv LConst$ . In the second but last step we use that $LConst$, being a left exact functor, preserves terminal objects. The last step is the definition (1) of the pro-left adjoint.
If $\mathbf{H}$ is locally ∞-connected, in that $\LConst$ has an actual left $(\infty,1)$-functor $Shp$ (see at shape modality, in constrast to just a pro-left adjoint (1)) then the shape of $\mathbf{H}$ is $Shp(*) \in \infty Grpd \xhookrightarrow Prop \infty Grpd$ (under the embedding here):
This follows immediately from Prop. and the observation that an actual left adjoint, when it exists, of coincides with the pro-left adjoint under the embedding $\infty Grpd \hookrightarrow Pro(\infty Grpd)$ (here)
But we may also see explicitly that we have the following sequence of natural equivalences of ∞-groupoids, starting with Def. (alternatively: Prop. (.
Thus, if we regard $Shp(*)$ 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 $X$, $Shp_{\mathbf{H}}(*)$ 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.
More generally:
Let $(f_! \dashv f^* \dashv f_*) : \mathbf{H} \xrightarrow{\;\;} \mathbf{B}$ 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 $(\infty,1)$-category of Grothendieck $(\infty,1)$-toposes and geometric morphisms (see here), we have
(cohesive (∞,1)-topos has trivial shape)
Every
$\infty$-topos which is both locally ∞-connected and $\infty$-connected,
in particular 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 $Shp$ preserves the terminal object, and apply the above observation that the shape of a locally ∞-connected (∞,1)-topos is represented by $Shp(*)$.
(trivial shape of gros $\infty$-toposes)
That cohesive $(\infty,1)$-toposes have trivial shape (Exp. ) is a reflection of their characteristic nature as gros toposes: Rather than representing a single specific non-trivial space, cohesive $\infty$-toposes are gros $\infty$-categories of spaces (of geometric/cohesive spaces, specifically).
This is further brought out by Prop. below, in view of which Exp. , may be read as saying that the shape of a cohesive $\infty$-topos does not interfere with the cohesive shapes of its objects.
If $\mathbf{H}$ is a cohesive $(\infty,1)$-topos with shape modality
then for every object $X \,\in\, \mathbf{H}$ the shape of the slice $(\infty,1)$-topos $\mathbf{H}_{/X}$, according to Def. , is equivalently the cohesive shape of $X$:
For $\mathbf{H}$ any $(\infty,1)$-topos and $X \,\in\, \mathbf{H}$ an object, the slice $(\infty,1)$-topos $\mathbf{H}_{/X}$ is related to $\mathbf{H}$ by the base change adjoint triple shown on the left here, together with, on the right, part of the adjoint quadruple that exhibits the cohesion of $\mathbf{H}$:
By essential uniqueness of adjoint $(\infty,1)$-functors (here) and of the terminal $(\infty,1)$-geometric morphism (here), the composite adjunction is the global section geometric morphism of the slice topos:
Hence, by Prop. , we need to exhibit a natural equivalence of this form:
Therefore, consider, for $S_1, S_2 \,\in\, Grpd_\infty$, the following sequence of natural equivalences:
Here the first four steps use the hom-equivalences of the above adjunctions. The second but last step uses cohesion, namely that the shape modality preserves finite homotopy products and is idempotent. The last step is the hom-equivalence for the cartesian monoidal-structure of ∞Grpd.
By the $(\infty,1)$-Yoneda lemma, this implies the equivalence (2) and hence the claim to be proven.
To any topological space $X$ is associated the its $\infty$-category of $\infty$-sheaves (with respect its site of open subsets), which is an $\infty$-topos.
At least when $X$ is a compact Hausdorff space, then its strong shape in the classical sense of Mardešić & Segal 1971 does agree with the $\infty$-topos theoretic shape of its $\infty$-category of $\infty$-sheaves.
This fact must have motivated the terminology in Toën-Vezzosi 2002 and in Lurie 2009, Sec. 7.1.6; it is made explicit in Hoyois 2013, Rem. 2.13.
The notion of shape of an $\infty$-topos first appears, in essentially the form of the above Def. , in:
The concise formulation as $\Gamma \circ LConst(-)$, as in the above Def. , and discussion of relation to classical (strong) shape theory, of topological spaces, is due to
The further re-formulation as the image of the terminal object under the pro-left adjoint to $LConst$ is highlighted in
See also
Last revised on October 9, 2021 at 18:56:34. See the history of this page for a list of all contributions to it.