nLab condensed infinity-groupoid

Idea

The original motivation behind condensed infinity-groupoids is to create well-behaved categories of mathematical structures such as condensed HZ-module spectra in which one could do derived analytic geometry using category-theoretic methods without resorting to not-so-well behaved categories of topological spaces.

However, David Corfield offers another motivation for condensed infinity-groupoids in the blog post Pyknoticity vs Cohesiveness on the nCafé, as a way to extend cohesion to the $p$-adic case (see at condensed cohesion):

Mathematics rather than physics is the subject of chapter 5 of my book, where I’m presenting cohesive HoTT as a means to gain some kind of conceptual traction over the vast terrain that is modern geometry. However I’m aware that there are some apparent limitations, problems with ‘$p$-adic’ forms of cohesion, cohesion in algebraic geometry, and so on. In the briefest note (p. 158) I mention the closely related pyknotic and condensed approaches of, respectively, (Barwick and Haine) and (Clausen and Scholze).

 Definition

A condensed infinity-groupoid or condensed anima is a hypercomplete (infinity,1)-sheaf of infinity-groupoids on the pro-étale (infinity,1)-site of the point, small relative to a universe $\mathcal{U}$.

Equivalently, it is an (infinity,1)-functor

$$T:ProfiniteSpaces^\op \longrightarrow \infty Grpd_\mathcal{U}$$

from the opposite (infinity,1)-category of profinite spaces to the (infinity,1)-category Infinity-Grpd of infinity-groupoids which are small relative to a universe $\mathcal{U}$, such that the natural maps

$$T(\mathbb{0}) \to \mathbb{1}$$

and

$$T(S + S^{'}) \to T(S) \times T(S^{'})$$

are homotopy equivalences for any profinite space $S$ and $S^{'}$, and the natural map

$$T(S) \to \operatorname{lim}_{\Delta} T(S_\bullet)$$

is a homotopy equivalence for any hypercover of profinite spaces $S_\bullet \to S$.

When restricted to the site of extremally disconnected compact Hausdorff spaces, hyperdescent is automatic, so the simplest equivalent definition of a condensed infinity-groupoid is simply a finite-product-preserving (infinity,1)-functor $T:ExtrDisc^\op \longrightarrow \infty Grpd_\mathcal{U}$, see (Barwick-Haine, corollary 2.4.4).

Properties

The (infinity,1)-category $\mathrm{Cond}\infty\mathrm{Grpd}$ of condensed infinity-groupoids is a locally small locally cartesian closed (infinity,1)-pretopos.

Peter Scholze speculates in this comment on the nCafé that $\mathrm{Cond}\infty\mathrm{Grpd}$ seem to form a predicative elementary (infinity,1)-topos. If true, then $\mathrm{Cond}\infty\mathrm{Grpd}$ is an example of an elementary (infinity,1)-topos which is not a Grothendieck (infinity,1)-topos.

Terminology

In the model of infinity-groupoids as simplicial sets/Kan complexes, condensed infinity-groupoids are sometimes referred to as condensed anima.

Related concepts

• condensed mathematics
• condensed set
• condensed spectrum
• condensed E-infinity ring
• condensed module spectrum

References

• David Corfield, Modal homotopy type theory, Oxford University Press 2020 (ISBN: 9780198853404)

• Clark Barwick, Peter Haine, Pyknotic objects, I. Basic notions, (arXiv:1904.09966

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

See condensed mathematics and infinity-groupoid#Anima.