nLab condensed local contractibility

Idea

The typical gros -toposes and -$\infty$-toposes considered in arithmetic/algebraic geometry are far from being cohesive over Set and even farther from being $\infty$-cohesive over $Grpd_\infty$, respectively: The extra left adjoint to their terminal geometric morphism would mean that varieties/schemes were locally connected or even locally $\infty$-connected, which they are not.

(This is in contrast to the situation in differential geometry, where smooth manifolds, formal smooth manifolds, and supermanifolds, etc., generate the cohesive $\infty$-toposes over $Grpd_\infty$ of smooth $\infty$-groupoids, formal smooth $\infty$-groupoids and super formal smooth $\infty$-groupoids, etc.)

But cohesion may usefully be considered over non-trivial base toposes or base $\infty$-toposes. For example, the slice $\infty$-topos of global equivariant homotopy theory over the archetypical $G$-orbi-singularity is cohesive over proper $G$-equivariant homotopy theory (see at cohesion of global- over G-equivariant homotopy theory).

Analogously, the gros topos of pro-étale arithmetic/algebraic geometry, while not cohesive over sets, has at least an extra left adjoint over the base topos of condensed sets. This and the analogous statement for the respective hypercomplete $\infty$-toposes is claimed in Scholze 2020, see also Scholze a.

In view of the terminology of “condensed mathematics” one might call this local contractibility of pro-étale arithmetic geometry relative to condensed sets: condensed local contractibility.

Statement

under construction

Definition

Given

• $k$ a separably closed field,

• $\kappa$ an uncountable strong limit cardinal

write

• $Sch^{\leq \kappa}_{/Spec(k)}$ for the category of $\kappa$-small schemes over $Spec(k)$,

• $Sch^{\leq \kappa}_{/_{pet} Spec(k)}$ for its full subcategory of those that are pro-étale over $Spec(k)$,

  equivalently this is the category of $\kappa$-small profinite sets,

both regarded as sites via the pro-étale topology, and write

(1)$$Sh\big( Sch^{\leq \kappa}_{/Spec(k)}\big) \underoverset {\underset{\Gamma}{\longrightarrow}} {\overset{LConst}{\longleftarrow}} {\;\; \bot \;\;} Sh\big( Sch^{\leq \kappa}_{/_{pet}Spec(k)}\big)$$

for the canonical geometric morphism between their sheaf toposes.

Remark

The base topos in (1) is also called the category of $\kappa$-condensed sets, or of $\kappa$-pyknotic sets (with $\kappa$ often omitted in the latter case).

The following seems to be claimed:

Proposition

The geometric morphism (1) has the following properties:

• $LConst$ is fully faithful;

• $\Gamma$ is a left adjoint

and

• $LConst$ is a right adjoint.

The first two claims are fairly standard, the last one would make the topos of pro-étale schemes be locally connected and hence almost cohesive over $\kappa$-condensed sets/$k$-pyknotic sets (by Rem. ).

Structures in condensed local contractibility

The point of axiomatic cohesion is that it formally implies the presence of structures of the form of key phenomena known in traditional differential topology, but now internal into any given cohesive topos/$\infty$-topos. Hence to the extent that pro-étale arithmetic geometry is cohesive over the pro-étale base topos it inherits these structures.

There would be much to be discussed here. The following lists some first observations with links to further commentary.

Differential cohomology diagram

Every spectrum object in a locally $\infty$-connected $\infty$-topos sits in the center of a differential cohomology diagram, which may be understood as decomposing the object into its various modal aspects, such that, at least for smooth $\infty$-groupoids, it exhibits each object as representing a Whitehead-generalized differential cohomology theory.

Since the construction of the differential cohomology hexagon does not require the extra left adjoint to preserve finite products, it should exist in pro-étale arithmetic geometry. Commentary in this direction is in Scholze b.

References

The idea of the possibility of pro-étale local contractibility is voiced in discussion here:

• Marc Hoyois, Urs Schreiber, étale ∞-site (around nForum comment 43431), November 2013

The claim of cohesion-except-for-product-preservation of gros pro-étale toposes over the corresponding pro-étale base topos appears in:

• Peter Scholze, Re: Pyknoticity versus Cohesiveness, March 2020

Related remarks and discussion:

• Peter Scholze, Answer to ‘What is the precise relationship between pyknoticity and cohesiveness?’, April 2020 (MO:https://mathoverflow.net/a/356836).

• Peter Scholze, Answer to ‘Cohesion relative to a pyknotic/condensed base’, Feb 2021 (MO:a/384900).

• Peter Scholze, Answer to ‘Condensed / pyknotic approach to orbifolds?’, Feb 2021 (MO:a/384898)

More concrete discussion, phrased in terms of fractured $(\infty,1)$-toposes:

• Qi Zhu, Fractured Structure on Condensed Anima, MSc thesis, Bonn (2023) [pdf, pdf]

exposition:

• Qi Zhu, Fractured structure on condensed spaces, talk notes (2023) [pdf, pdf]

• Nima Rasekh, What is a topological structure?, talk notes (April 2023) [pdf, pdf]