# Questions tagged [condensed-mathematics]

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

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### 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 ...

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### 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{...

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### 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 ...

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### 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 ...

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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 ...

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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}} \...

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### 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 ...

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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{...

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### 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 ...

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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? ...

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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 ...

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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 ...

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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 ...

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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:= \...

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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 ...

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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 ...

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### 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 ...

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### 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 $\mathbb{F_p}$-...

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### 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=...

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### 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 ...

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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 $...

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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 ...

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### 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, ...

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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 ...

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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 ...

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### 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 ...

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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....

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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 ...

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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 (...

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### 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 ...

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### 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 ...

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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 ...

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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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### 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 ...

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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],\...

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### 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 ...

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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 ...

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### 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-...

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### 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 ...

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### Topological localisations of algebras and solidification

Let $A$ be a (topologically discrete) commutative ring and consider the topological ring $A((z))$. Let $\underline{A((z))}$ denote the corresponding condensed ring, and let $a(z)$ be a non-zero ...

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### Reflexive double dual in condensed math

Consider the site of profinite sets $\mathcal{S}$. In condensed math we consider sheaves of abelian groups $C = \text{Sh}(\mathcal{S}, \text{Ab})$ on this site. It has a natural tensor product and ...

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### Solid modules and algebraic loop spaces

Background. Let $X$ be an affine variety over a field $k$ of characteristic $0$. We can define then the loop space of $X$, denoted $\mathcal{L}X$, to be the functor taking a $k$-algebra $A$ to the set ...

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### Solid tensor product of pro-discrete space with Laurent series

Consider the category $\operatorname{Solid}_{\mathbf{Z}}$ of solid abelian groups in the sense of Clausen-Scholze. This category is a full subcategory of condensed abelian groups, $\operatorname{Cond}...

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### Condensed math and cofiltered limits

I have a question involving preservation of cofiltered limits. Ordinarily this would be a very boring question, but it comes up in condensed math in its analogue of the completeness concept.
The ...

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### Homotopy theory with condensed sets

Is there a canonical way for doing homotopy theory with condensed sets? Is there a definition of homotopy groups? As CW complexes are compactly generated Hausdorff we can consider then as condensed ...

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### Appropriate notion of derived category over condensed set

If we have a compact Hausdorff space $S$, then my understanding is that the appropriate notion of the derived category of sheaves of condensed abelian groups is to consider the derived category $D_{\...

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### Difference between definitions of continuous action, profinite case

My setting is the following : let $G$ be a topological group and $X$ be a topological space. I have the head filled with two possible definitions for a continuous action of $G$ on $X$.
The first could ...

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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 ...

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### The site of extremally disconnected sets

In proposition 2.7. of the condensed notes of professors Scholze and Clausen it is said that the category of extremally disconnected sets is a site, but in the definition of a site in the Stacks ...

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### Limits and colimits in the category of condensed abelian groups

Sheafification is needed in limits and colimits of condensed abelian groups? If I have a functor $T: i \mapsto T_i$ from an index category to condensed abelian groups the limit and colimit of this ...