Questions tagged [condensed-mathematics]
Condensed mathematics of Clausen and Scholze. Closely related to the pyknotic mathematics of Barwick and Haine.
112 questions
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Hölder's inequality for trace-class maps of $p$-liquid spaces and a related conjecture of Grothendieck
In Condensed Math and Complex Geometry Proposition 8.8, Clausen-Scholze describe trace-class maps between projective objects in the $p$-liquid category as sums of rank 1 operators against ${<}p$-...
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What intuitive notion is formalized by condensed mathematics?
Preface: I ask this question from the position of a curious layperson who is excited about new conceptual advances in mathematics.
It is often said that the notion of "topology" formalises ...
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Left Adjoint From the Category of Topological Groups to the Category of Condensed Groups
In Scholze's Lecture Notes on Condensed Sets, the author states that (Remark 1.8) the functor that takes a topological group $G$ to its condensation $\underline{G}$ has a left-adjoint, but we do not ...
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What remains true after condensation?
As I slowly develop intuition for the condensed formalism, I feel that it’d help greatly if there were a principle or meta-theorem which said that certain kinds of statements that are true in ordinary ...
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The Frobenius and Dold–Puppe
At the end of Lecture 12 in the Clausen–Scholze, Lectures on Analytic Geometry, there is a discussion of the exact endofunctor of $D_{\geq 0}(A)$ defined by $Q\mapsto \operatorname{colim}_n \...
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Can you glue the Betti $X_\text B$ and de Rham stacks $X_\text{dR}$ together?
Let $X$ be a complex algebraic variety. Is there a (derived prestack) over a base
$$\pi\ :\ X_\text{dR,B}\ \to\ S$$
where $S=\mathbf{R},\mathbf{R}/\mathbf{R}^\times,\mathbf{A}^1_\mathbf{C}/\mathbf{G}...
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Construction of exceptional functors $f_!$ (and $f^!$) when not finitely generated over $\mathbb{Z}$
Clausen and Scholze introduced the concept of an analytic ring, leading to the following results:
Theorem (Clausen-Scholze)
(1) If $R$ is a finitely generatedalgebra over $\mathbb{Z}$, then the pre-...
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Stalks of nonarchimedean spaces as analytic rings
Let $(A,A^+)$ be an affinoid Tate ring, and let $x \in X=\operatorname{Spa}(A,A^+)$. When defining the stalks of the structure sheafs ${\mathcal O}_{X,x} = \varinjlim_{x \in U} {\mathcal O}_{X}(U) $ ...
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Are dualizable topological vector spaces finite-dimensional?
Consider the symmetric monoidal category TVS of complete Hausdorff topological vector spaces equipped with the completed projective, injective, or inductive tensor product.
Every finite-dimensional ...
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What does the topos of (light) condensed sets classify?
Recall that $\mathrm{Pro}(\mathbf{FinSet}) = *_{\text{proét}}$, the category of profinite sets, forms a site with finite jointly surjective families as covers, and that the category of sheaves on this ...
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"Noetherianess" of $\mathrm{Mod}(\mathbb{F}_{p,\square})$
In classical commutative ring theory it is quite immediate to see, that a field is noetherian in the following sense:
For any finitely generated $k$-vectorspace $M$, any sub-object is finitely ...
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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 ...
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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 ...
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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?
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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 ...
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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 ...
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$\mathbb{Z}[T]$-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 ...
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"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 ...
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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 $(\:\underline{Y}\,)^X$ ...
user103549
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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 ...
user473423
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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 ...
user473423
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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, ...
user473423
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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 ...
user473423
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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 (...
user473423
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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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