(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 -topos is the underlying pro-homotopy type of this space (Def. below). Equivalently, the shape of an -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 in the classical sense of shape theory.
In the special case that the -topos is a slice of a cohesive -topos 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 -topos – Toën-Vezzosi 2002, Def. 5.3.2)
The composite (∞,1)-functor
is the shape functor . Its value
on an -topos is the shape of .
Here:
(∞,1)Topos is the (∞,1)-category of (∞,1)-toposes;
∞Grpd is the -category of ∞-groupoids;
is the (∞,1)-Yoneda embedding;
is the (∞,1)-category of (∞,1)-functors;
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.
is the functor that produces the (∞,1)-category of (∞,1)-presheaves on (equivalently on the equivalent opposite ∞-groupoid );
is the (∞,1)-category of pro-objects in .
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 there is a unique geometric morphism
where
∞Grpd is the -topos of ∞-groupoids
is the global sections (∞,1)-functor
is the constant ∞-stack functor.
(shape of an -topos – Lurie 2009, Def. 7.1.6.3)
The shape of an -topos is the composite -functor
For ∞Grpd we have by the (∞,1)-Grothendieck construction-theorem and using that up to equivalence every morphism of -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 is the canonical projection . This means that it is an etale geometric morphism. So for any geometric morphism 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 there is a canonical morphism
The image of this under is (using that this preserves the terminal object) a morphism in of the form
Conversely, given a morphism of the form in we obtain the base change geometric morphism of slice -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 according to Def. does preserve finite -limits, since in the equivalent Def. this is manifest: There clearly preserves all limits, since it is a right adjoint, and preserves finite limits, since it is a left exact functor by definition. Similarly, this makes manifest that is accessible, since and are both accessible.
(shape of an -topos – Hoyois 2013, p. 3)
For an -topos, write (with convenient overloading of notation)
for the pro-left adjoint to and say that the shape of is the image under this pro-left adjoint of its terminal object :
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 . In the second but last step we use that , being a left exact functor, preserves terminal objects. The last step is the definition (1) of the pro-left adjoint.
If is locally ∞-connected, in that has an actual left -functor (see at shape modality, in constrast to just a pro-left adjoint (1)) then the shape of is (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 (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 as “the fundamental ∞-groupoid of ” — which is reasonable since when consists of sheaves on a locally contractible topological space , is equivalent to the usual fundamental ∞-groupoid of — then we can regard the shape of an -topos as a generalized version of the “homotopy -groupoid” which nevertheless makes sense even for non-locally-contractible toposes, by taking values in the larger category of “pro--groupoids.”
It follows also that is not only locally ∞-connected but also ∞-connected, then it has the shape of a point.
More generally:
Let be an essential geometric morphism of -toposes that exhibits as an essential retract of in that
Then the shape of is equivalent to that of .
Since is the terminal object in the -category of Grothendieck -toposes and geometric morphisms (see here), we have
(cohesive (∞,1)-topos has trivial shape)
Every
-topos which is both locally ∞-connected and -connected,
in particular every cohesive (∞,1)-topos
over has the shape of the point.
By definition is -connected if the constant ∞-stack inverse image is
not only a left but also a right adjoint;
By standard properties of adjoint (∞,1)-functors we have that a right adjoint is a full and faithful (∞,1)-functor precisely if the counit is an equivalence.
Equivalently, we can observe that a locally ∞-connected (∞,1)-topos is ∞-connected precisely when preserves the terminal object, and apply the above observation that the shape of a locally ∞-connected (∞,1)-topos is represented by .
(trivial shape of gros -toposes)
That cohesive -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 -toposes are gros -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 -topos does not interfere with the cohesive shapes of its objects.
If is a cohesive -topos with shape modality
then for every object the shape of the slice -topos , according to Def. , is equivalently the cohesive shape of :
For any -topos and an object, the slice -topos is related to 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 :
By essential uniqueness of adjoint -functors (here) and of the terminal -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 , 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 -Yoneda lemma, this implies the equivalence (2) and hence the claim to be proven.
To any topological space is associated the its -category of -sheaves (with respect its site of open subsets), which is an -topos.
At least when is a compact Hausdorff space, then its strong shape in the classical sense of Mardešić & Segal 1971 does agree with the -topos theoretic shape of its -category of -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 -topos first appears, in essentially the form of the above Def. , in:
The concise formulation as , 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 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.