nLab condensed infinity-groupoid



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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 pp-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 ‘pp-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).


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 𝒰\mathcal{U}.

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

T:ProfiniteSpaces opGrpd 𝒰T: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 𝒰\mathcal{U}, such that the natural maps

T(𝟘)𝟙T(\mathbb{0}) \to \mathbb{1}


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

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

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

is a homotopy equivalence for any hypercover of profinite spaces S SS_\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 opGrpd 𝒰T:ExtrDisc^\op \longrightarrow \infty Grpd_\mathcal{U}, see (Barwick-Haine, corollary 2.4.4).


The (infinity,1)-category CondGrpd\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 CondGrpd\mathrm{Cond}\infty\mathrm{Grpd} seem to form a predicative elementary (infinity,1)-topos. If true, then CondGrpd\mathrm{Cond}\infty\mathrm{Grpd} is an example of an elementary (infinity,1)-topos which is not a Grothendieck (infinity,1)-topos.


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


See condensed mathematics and infinity-groupoid#Anima.

Last revised on July 29, 2022 at 17:02:27. See the history of this page for a list of all contributions to it.