# nLab condensed cohesion

Contents

Speculative page under construction

# Contents

## Idea

Notions of cohesion are generally treated with ∞Grpd as the base (∞,1)-topos. However, for the purposes of algebraic geometry and arithmetic geometry, the (∞,1)-topos of (∞,1)-sheaves over the pro-étale site of a point, $Sh_\infty(ProFinSet)$, may be better suited.

Since this site is used centrally in condensed mathematics, we might call cohesion over this new base, condensed cohesion. On the other hand, it is the closed related pyknotic approach that constructs an (∞,1)-topos of pyknotic spaces.

## Infinitesimal cohesion

The tangent (∞,1)-topos of $Sh_\infty(ProFinSet)$ ought to be composed of some kind of parameterized spectra. (Pyknotic spectra form the stabilization of the $(\infty, 1)$-category of pyknotic spaces (BarHai19, sec 3.1).)

## Cohesion

The big pro-étale site on all schemes over a separably closed field $k$ is almost (check) cohesive over the pro-étale site of a point.

Consider then the construction in cohesive (infinity,1)-topos – structures that do not rely on finite-product preservation in the left adjoint.

The differential cohomology diagram for stable objects in sheaves over the former…

## References

Last revised on February 22, 2021 at 15:23:55. See the history of this page for a list of all contributions to it.