Questions tagged [condensed-mathematics]

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

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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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2 votes
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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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5 votes
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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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2 votes
1 answer
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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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17 votes
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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 ...
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Are condensed vector spaces over finite fields always solid?

The Clausen-Scholze theory of condensed mathematics offers an abelian category with enough projective objects that embraces the study of arbitrary locally compact (and Hausdorff) groups. The behaviour ...
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8 votes
1 answer
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Compactly supported sections of coherent sheaves and the dualizing complex

Suppose $U$ is a (possibly singular) scheme and $X$ is a compactification (potentially unnecessary at least in characteristic $0$). Let $\pi:X\to *$ be the map to the point (though one can consider ...
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7 votes
1 answer
536 views

Relation between ProCoh and solid modules

There are two languages endow the theory of coherent sheaves with a six functor formalism (that I "know" of), one being formulated in $\text{ProCoh}(X)$ by Deligne and the other being $D(\...
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12 votes
3 answers
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Witt vectors, the cotangent complex, and a solid construction of $B_{dR}^+$

In a remarkable lecture delivered on October 29th: New Foundations for functional analysis, Dustin Clausen suggests at the 40 minute mark a remarkable new construction interpretation of Fontaine's ...
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Condensed/liquid vector spaces and path integrals

[Edited to take into account comments.] Background One approach to the problem of making rigorous various measures on spaces of paths (for example, the Wiener or Feynman measure) is the time-slicing ...
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12 votes
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Condensed mathematics and independence results

I recently saw a paper on ``condensed mathematics'', in which I found the following quote interesting (see Condensed Mathematics: The internal Hom of condensed sets and condensed abelian groups and a ...
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16 votes
1 answer
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Examples of solid abelian groups

I am reading through Clausen's and Scholze's Lectures on condensed mathematics. I am struggling to understand the concept of solid abelian groups so I am looking for some examples. Is the underlying ...
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Reconstructing an analytic ring from its module category

When reading Lectures on Analytic Geometry, I found that in the data of an analytic ring, the underlying ring seems unnatural and the module category should be the soul, but in fact one needs the ...
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An attempt to define partial properness and compactification for some maps between analytic spaces

The paper Étale cohomology of diamonds defines partial properness and compactification for maps between v-sheaves, and in particular for perfectoid spaces and rigid-analytic spaces. Recently when ...
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11 votes
1 answer
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Are the “topologies” arising from constructive type theories with quotients actually condensed sets?

This is the second in a pair of questions. For the other see Are representations in computable analysis the equivalent to countably-generated condensed sets?. Dustin Clausen and Peter Scholze have a ...
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Are representations in computable analysis the equivalent to countably-generated condensed sets?

This is the first in a pair of questions. For the other see here. Dustin Clausen and Peter Scholze have a theory of condensed sets, which is a slightly different take on topology. For most cases, ...
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15 votes
2 answers
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What are the points (and generalized points) of the topos of condensed sets?

What are the topos points of $\mathrm{CondSet}$, i.e. the geometric morphisms $\mathrm{Set} \to \mathrm{CondSet}$? More generally, is there a concise description of the geometric morphisms $\mathcal{E}...
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8 votes
0 answers
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Nonabelian variants of the Breen-Deligne resolution

The Breen-Deligne resolution is an unusual functorial resolution of an abelian group A by finite direct sums of free abelian groups of the form $\Bbb Z[A^n] = Free_{Ab}(A^n)$. It makes several ...
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Pseudocoherent analogue of compact + nuclear = dualizable?

$\DeclareMathOperator\RHom{RHom}\DeclareMathOperator\Map{Map}\DeclareMathOperator\id{id}\DeclareMathOperator\colim{colim}$Let $(\mathcal A,\mathcal M)$ be a (normalized) analytic ring defined in ...
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Non-induced analytic structures in complex-analytic case

In Lectures on Analytic Geometry, for complex-analytic geometry, seemingly one only considers maps $(\mathbb C,\mathcal M_{<p})\to(\mathcal A,\mathcal M)$ of analytic rings for $0<p\le1$ where $...
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Flatness of maps of analytic rings

Reference: Lectures on Analytic Geometry Let $f\colon(\mathcal A,\mathcal M)\to(\mathcal B,\mathcal N)$ be a map of analytic ring. There are several possible ways to pose the flatness: Flatness as ...
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Analytic structures on the source of a surjection of condensed rings

Question. Let $(\mathcal B,\mathcal N)$ be an analytic (animated associative) ring, $\mathcal A$ be a condensed (animated associative) ring and $f\colon\mathcal A\to\mathcal B$ a surjective map of ...
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6 votes
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Algebraic K-theory "with proper support"

I would like to know what is the "correct" algebraic $K$-theory "with proper support". I suppose that the answer should be found in the condensed world, which is mostly inspired ...
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8 votes
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Derived category of abelian sheaves on a site equivalent to sheaves on the derived category of abelian groups

Reading Scholze's notes on Condensed Mathematics it is mentioned that when considered as $\infty$-categories, $$ D(\operatorname{Cond(Ab)}) \cong \operatorname{Cond}(D(\operatorname{Ab}))$$ and that ...
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13 votes
1 answer
954 views

$\infty$-topoi versus condensed anima

Let $ExDisc_\kappa$ denote the category of $\kappa$-small extremally disconnected topological spaces (for now fix a strong limit cardinal $\kappa$). There's a functor $ExDisc_\kappa \to \mathsf{RTop}$ ...
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3 votes
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Countable sum $\bigoplus_{n=0}^\infty\mathbb Z_p$ as a topological group

$\DeclareMathOperator\colim{colim}$This is inspired by Clausen's answer. Question: Recall that $\mathbb Z_p$ is endowed with the $p$-adic topology. Consider the countable sum $M:=\bigoplus_{n=0}^\...
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10 votes
2 answers
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Uniform spaces as condensed sets

$\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\Unif{Unif}\DeclareMathOperator\CHaus{CHaus}\DeclareMathOperator\Set{Set}\DeclareMathOperator\op{op}\DeclareMathOperator\Ind{Ind}\DeclareMathOperator\...
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1 vote
1 answer
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Are absolute Galois groups condensed?

Let $k^{s}$ be a separable closure of a field $k$. Is $Gal(k^s/k)$ a condensed group in the sense of condensed mathematics? If condensed, is it always solid?
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62 votes
2 answers
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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(\...
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14 votes
2 answers
2k views

Condensed / pyknotic sets in terms of forcing over Boolean-valued models of set theory / multiverse concepts?

Here is one way of saying what a pyknotic set is. Fix an inaccessible cardinal $\kappa$, and let $Proj_\kappa$ be the category of $\kappa$-small, extremally disconnected compact Hausdorff spaces. ...
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7 votes
1 answer
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Solidification of free abelian group on compact Hausdorff space

In the lecture notes on condensed mathematics the solidification of the free condensed abelian group $\mathbb{Z}[S]$ on a profinite set $S$ is defined as the inverse limit $\lim_{\leftarrow} \mathbb{Z}...
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3 votes
1 answer
551 views

Duality between $D^b(\mathbb{Z})$ and $D(\mathrm{Solid})^\omega$

My question is about Corollary 6.1(ii) in Lectures on Condensed Mathematics by Scholze (page 41). Here is the claim: The derived category $D(\mathrm{Solid})$ is compactly generated, and the full ...
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9 votes
1 answer
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Cohesion relative to a pyknotic/condensed base

Something that usefully emerged for me from this discussion and follow-up MO question is that rather than see cohesiveness and condensedness/pyknoticity in rivalry with one another, as my initial ...
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9 votes
1 answer
925 views

Condensed / pyknotic approach to orbifolds?

Does condensed / pyknotic mathematics afford an (yet!) another approach to orbifold theory? Let me admit up-front that I don't know much about either condensed / pyknotic mathematics or about orbifold ...
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14 votes
1 answer
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Is there a condensed / pyknotic refinement of the shape of an $\infty$-topos?

Let $\mathcal E$ be an $\infty$-topos. Recall that Lurie defines the shape of $\mathcal E$ as the left-exact, accessible functor $\Gamma \Delta: Spaces \to Spaces$ where $\Delta: Spaces^\to_\leftarrow ...
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1 vote
1 answer
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When is a complex of Banach spaces exact as condensed abelian groups?

Lectures on Condensed Mathematics, Theorem 3.3 says that for any compact Hausdorff space $S$, the cohomology $H_{\mathrm{cond}}^i(S,\mathbb R)=0$ for $i>0$ and $H_{\mathrm{cond}}^0(S,\mathbb R)=C(S,...
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22 votes
1 answer
2k views

Condensed criterion for sheafiness of adic spaces

Multiple times in talks about condensed mathematics (e.g. the Masterclass talks, Clausen's RAMpAGe talk), it is stated that the derived structure sheaf given by the condensed formalism "fixes&...
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3 votes
1 answer
380 views

Question about adjoint of forgetful functor from condensed abelian groups to condensed sets

There is a forgetful functor from condensed abelian groups to condensed sets. According to Scholze's notes, this has an adjoint $T \mapsto \mathbb{Z}[T]$ (which is the sheafification of the functor ...
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