Questions tagged [condensed-mathematics]

Condensed mathematics of Clausen and Scholze. Closely related to the pyknotic mathematics of Barwick and Haine.

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Analogue of $\ell^2(X)$ over an arbitrary Banach ring

Let $X$ be a set. Over the Banach fields $F=\mathbb{R}$ or $F=\mathbb{C}$ we can define the Banach space$$\ell^2(X)=\{\xi\colon X\to F\mid \sum_{x\in X}|\xi(x)|^2<\infty\}$$which satisfies a list ...
35 votes
1 answer
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Clausen–Scholze's Theorem 9.1 of Analytic.pdf, in view of light condensed sets, AKA is the Liquid Tensor Experiment easier now?

In the recent lecture series run jointly from IHÉS and Bonn, Clausen and Scholze have reworked—again—their foundations of geometry to focus attention on not arbitrary condensed sets and solid modules ...
3 votes
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Analytic number theory and condensed mathematics

As of 2024, are there current or planned applications of condensed mathematics to analytic number theory? If so, what are suggested readings?
67 votes
3 answers
11k views

Nonconvexity and discretization

Edit: Here's a more down-to-earth, and somewhat weakened, but I believe still nontrivial, version of the main theorem. Prototypical nonconvex spaces are $\ell^p$-spaces for $0<p<1$, say $\ell^p(\...
11 votes
1 answer
581 views

On the classification of second-countable Stone spaces

Let $X$ be a Stone space (i.e. totally disconnected compact Hausdorff). Then the following are equivalent: $X$ is second countable $X$ is metrizable $X$ has countably many clopen subsets $X$ is an ...
8 votes
1 answer
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On the connections between condensed mathematics and homotopy theory

I have a few questions, but they're not properly formulated just yet, but they stem from a few simple facts : In homotopy theory, the homotopy hypothesis postulates that topological spaces (up to ...
4 votes
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535 views

$\mathbb{Z}[T]$-Solidification in light condensed setting

In the lectures to "Analytic Stacks" Scholze and Clausen introduced a new concept of "light" condensed mathematics. In Lecture 7 Clausen introduces the derived $T$-solidification ...
23 votes
1 answer
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Condensed vs pyknotic vs consequential

As is probably clear from my previous questions, I am coming to "condensed mathematics" from the naive perspective of a category theorist, without much knowledge of the intended applications ...
29 votes
3 answers
4k views

What is the precise relationship between pyknoticity and cohesiveness?

Pyknotic and condensed sets have been introduced recently as a convenient framework for working with topological rings/algebras/groups/modules/etc. Recently there has been much (justified) excitement ...
3 votes
1 answer
515 views

Do CGWH spaces form an exponential ideal in Condensed Sets?

If $X$ is any condensed set and $Y$ is a compactly generated weak Hausdorff (CGWH) space (a.k.a. $k$-Hausdorff $k$-space), is $Y^X$ again a CGWH space? To be more precise, is $(\:\underline{Y}\,)^X$ ...
3 votes
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227 views

"weakly functorial resolution" of quasi-compact T_1 topological space by quasi-compact Hausdorff space

I have an arguably weird question: Let $X$ be a quasi-compact $T_1$ topological space, could there be a construction that takes such an $X$ as input and outputs a surjection $$X' \to X$$ with the ...
5 votes
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203 views

Duality and compactness for pro vector spaces

I have a somewhat basic question which I haven't been able to piece together from the literature. Background. We work over a field $\bf{k}$. Consider the category, $\bf{Pro}_{k}$, of pro- vector ...
8 votes
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387 views

Sheaf of compact Hausdorff spaces but not a condensed anima

Consider the site $\mathbf{CHaus}$ of compact Hausdorff spaces together with the finitely jointly surjective families of maps as coverings. Restriction induces an equivalence of categories $$ \mathbf{...
33 votes
1 answer
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What actually is the idea behind the condensed mathematics?

Condensed mathematics is the (potential) unification of various mathematical subfields, including topology, geometry, and number theory. It asserts that analogs in the individual fields are instead ...
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419 views

Local isomorphism of condensed sets and étale condensed groupoids

Is there a notion of local isomorphism for condensed sets? $\textbf{Motivation:}$ I am trying to define what an étale condensed groupoid would be. A topological groupoid $\mathcal{G}$ is said to be ...
5 votes
1 answer
698 views

Symmetric powers, localisation and Frobenius

I am trying to understand the proof of lemma 2.2.18 in Lucas Mann's thesis. Its statement is surprising for me, because it talks about general rings which are not necessarily characteristic $p$, and ...
3 votes
1 answer
503 views

One-point compactification of a condensed set

Is there a notion of a 'one-point compactification of a condensed set'? $\textbf{Motivation:}$ For a locally compact space $X$, there is a notion of maps that vanish at infinity. A continuous function ...
17 votes
3 answers
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Reference request for condensed math

I am trying to follow the beautiful notes by Peter Scholze on condensed mathematics (https://www.math.uni-bonn.de/people/scholze/Condensed.pdf) I am noting that the hard time that I am getting is a ...
1 vote
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Two different definitions of condensed groupoid

I am searching for a condensed version of a topological groupoid and I found two possible definitions. $\textbf{Definition 0:}$ A condensed groupoid(0) is a functor $X: \mathrm{Extr}^{\mathrm{op}} \...
4 votes
1 answer
409 views

Group action on a condensed set and its orbit space

Let $X$ be a condensed set, and let $G$ be a (discrete) group. Suppose we have an action $G$ on $X$, which is a group morphism $a:G \rightarrow \mathrm{Aut}(X)$, where $\mathrm{Aut}(X)$ is the group ...
10 votes
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Noncommutative condensed sets

Ignoring set-theoretic problems, we can see condensed sets as sheaves of compact Hausdorff spaces. Using Gelfand Duality we obtain an equivalence of categories \begin{align*} \mathrm{CHaus}^{\mathrm{...
7 votes
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478 views

Condensed categories vs categories (co)tensored with condensed sets

I am not sure how to solve set-theoretic issues properly, so let me first ignore them. There are two notions, probably closely related: Condensed categories, i.e. condensed objects in the category of ...
5 votes
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Definition of condensed $C^{\ast}$-algebra

The classical definition of a $C^{\ast}$-algebra is a Banach algebra with an isometric antilinear involution map $a \mapsto a^\ast$. What would be a good definition for a condensed $C^{\ast}$-algebra? ...
12 votes
1 answer
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Are condensed sets (locally) cartesian closed?

The category of condensed sets is the colimit of the toposes of $\kappa$-condensed sets over all cardinals $\kappa$, or equivalently the category of "small sheaves" on the large site of all ...
12 votes
1 answer
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Properties of pyknotic sets

In Peter Johnstone's 1979 paper On a topological topos, he proposed the topos of sheaves on the full subcategory of topological spaces spanned by the single object $\mathbb{N}_\infty$, the one-point ...
7 votes
3 answers
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Condensed Pontryagin duality

Has Pontryagin duality been extended to condensed abelian groups? The obvious approach being to define $\hat M$ as the internal hom to the circle group. Is it true that $\hat{\hat M}=M$ with this ...
1 vote
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Dualizable object that is not discrete

In Example 9.6 of Clausen-Scholze's Condensed Mathematics and Complex Geometry, they give an example of a dualizable object that is not discrete. In the process of doing so, they define $V_0:= \...
19 votes
2 answers
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Breen-Deligne packages and the liquid tensor experiment

I am trying to understand some things about Condensed Mathematics and the Liquid Tensor Experiment. The aim of the LTE is to provide a formalised proof of Theorem 9.4 in Scholze's paper Lectures on ...
21 votes
1 answer
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Expected applications of condensed mathematics

As a student of algebraic geometry (in an advanced stage, but still far from an expert on anything), I am quite excited about learning some condensed mathematics. I have been told that the theory has ...
0 votes
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195 views

Double complex of simplicial resolution

In his lectures on condensed mathematics on page 30 Peter Scholze speaks of the double complex of a simplicial resolution. How is this defined? In the next line, he writes that if $A_\bullet$ is a ...
15 votes
1 answer
2k views

Is there a good theory of solid vector spaces?

Lately I have become interested in solid $F$-modules where $F$ is some discrete field. Ideally, one would want a category that is as nicely behaved as solid abelian groups or solid $\mathbb{F_p}$-...
8 votes
1 answer
873 views

A hypercover of profinite sets as a limit of hypercovers of finite sets

This is about a rather concrete problem that occurs in the middle of a lecture by Scholze. First I'll refer to the lecture, but then I'll state the problem. In https://www.youtube.com/watch?v=...
6 votes
1 answer
772 views

Domain of left adjoint from condensed sets to anima

$\DeclareMathOperator\Hom{Hom}$Let $X$ be a condensed set in the sense of Clausen-Scholze. If there is a universal anima $Y$ (or $\infty$ groupoid, or homotopy type) together with a map of condensed ...
4 votes
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Derived completeness and condensed mathematics

This is a vague question: Does condensed mathematics have something to do with the notion of derived completeness? Namely, for a ring $R$ and an ideal $I$, one can speak about the category of derived $...
19 votes
1 answer
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Derived categories and $\infty$-categories necessary for condensed mathematics

I am reading the three texts on condensed mathematics by Scholze and Clausen. I am also interested in paper "A $p$-adic 6-functor formalism in rigid-analytic geometry" by Lucas Mann. To ...
1 vote
1 answer
350 views

Isomorphism of RHoms in condensed mathematics

In Proposition 5.7 on page 34 in lectures on condensed mathematics Peter Scholze shows that $\mathbb{Z}[S]^\blacksquare$ is solid. He shows that the two relevant expressions are isomorphic, however, ...
2 votes
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The map $\mathbb{Z}[S] \rightarrow \lim_i \mathbb{Z}[S_i]$ is injective

In the proof of proposition 2.1. of Analytic.pdf there is the following map: Let $S = \lim_i S_i$ a profinite set. Let $p_i: S \rightarrow S_i$ be the projection. We can define the following map using ...
9 votes
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How much condensed mathematics can be founded on finite order arithmetic (or ETCS) instead of ZFC?

I recently learnt from David Roberts' answer that, there is a way, due to Colin McLarty, to set up the foundations on finite order arithmetic for EGA & SGA. In particular, all usages of ...
5 votes
1 answer
697 views

Hypercovers consisting of finite sets

In this paper on Page 21, the first line of the proof, Peter Scholze seems to claim that any hypercover, consisting of finite sets, splits. I find this hard to believe. I am not familiar with ...
26 votes
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Is there a proof of Hodge theory using condensed mathematics?

As is well known, many results in complex geometry "feel" algebraic (and often have statements which are "completely algebraic") but only have "transcendental" proofs (i....
19 votes
1 answer
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Mixing solids and liquids

Is there a nontrivial way to consider products of archimedean and non-archimedean spaces in the context of Clausen–Scholze's analytic geometry? Context: Last week during a conference in Essen (School ...
4 votes
1 answer
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Condensed mathematics

I have a little technical question on Peter Scholze's lectures on condensed mathematics. On page 12, right above the Proof of Theorem 2.2, he says that for extremally disconnected sets the condition (...
2 votes
1 answer
404 views

Possible characterisation of compactly generated weakly Hausdorff spaces

Is it true that, in the category $\mathbf{Top}$ of topological spaces and continuous maps, the compactly generated weakly Hausdorff spaces are precisely the spaces arising as filtered colimits of ...
5 votes
1 answer
650 views

Structure of a profinite group as a condensed set with an action of an open subgroup

Let $G$ be a profinite group and $H$ be an open subgroup. As a continuous $H$-topological space, we have $G=\coprod_{G/H} H$. Does this also hold as condensed sets, i.e. do we have an identification ...
8 votes
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Analytification of DG-categories over $\mathbb C$?

In recent notes of complex geometry by Clausen–Scholze, they gave a theory of analytification of finite type $\mathbb C$-schemes. It seems to me that there is a non-commutative analogue which works ...
3 votes
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Verify that $M \otimes^L_{(A,A^+)_\blacksquare} (A,A)_\blacksquare \in D(\text{Cond}(B))$ lies in $D(B,\widetilde{B^+[T]})_{\blacksquare}$

I am struggling with verifying a proof in Scholze's notes on condensed mathematics. Let $(B,B^+)$ be a discrete Huber pair and let $(A,A^+) = (\mathbb{Z}[T],\mathbb{Z})$ and $(A,A) = (\mathbb{Z}[T],\...
2 votes
1 answer
356 views

Groupoid of points, shape and stratified shape of $\operatorname{Sh} (X_\text{pro-ét})$

$\DeclareMathOperator\Sh{Sh}\DeclareMathOperator\Pt{Pt}$Maybe this is well-known or even a stupid misunderstanding of something very basic. It's well-known that the groupoid of points (i.e., groupoid ...
4 votes
1 answer
742 views

Adjunction between topological spaces and condensed sets

I am trying to prove that the functor \begin{align*} \mathrm{Top} &\longrightarrow \mathrm{Cond}(\mathrm{Set}) \\ X &\longmapsto \underline{X} \end{align*} admits a left adjoint and it is the ...
7 votes
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Algebraizing topology and analysis via condensed mathematics

I asked this question on Mathematics Stackexchange, but one of the users suggested that I ask this question at MathOverflow. I've just come across a Twitter thread by Laurent Fargues explaining a work ...
7 votes
1 answer
555 views

What is the reason for $f_!$ not preserving discrete objects?

Let $A$ be a finitely generated $\mathbb{Z}$-algebra and let $f: \operatorname{Spec} A \rightarrow \operatorname{Spec} \mathbb{Z}$ be the canonical map. On pg. 53, Thm. 8.2 of https://www.math.uni-...