# Questions tagged [condensed-mathematics]

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

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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 ...

**3**

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**1**answer

387 views

### 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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389 views

### 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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245 views

### 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 ...

**4**

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**1**answer

422 views

### 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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**1**answer

561 views

### 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 ...

**13**

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**1**answer

561 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 $ExDisc_\kappa \to \mathsf{RTop}$ ...

**3**

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468 views

### 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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890 views

### 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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674 views

### 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?

**52**

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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(\...

**14**

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**2**answers

1k 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. ...

**6**

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**1**answer

407 views

### 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}...

**3**

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**1**answer

351 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 ...

**9**

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**1**answer

956 views

### 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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716 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 ...

**13**

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538 views

### 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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293 views

### 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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1k 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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307 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 ...

**7**

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430 views

### Different definitions of condensed sets

The $\kappa$-condensed sets are defined as the sheaves on the site of profinite spaces of cardinality less than $\kappa$ (with $\kappa$ an uncountable strong limit cardinal) with morphisms the ...

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### Infinity-categorical analogue of compact Hausdorff

Recently I became through this mathoverflow question aware of the article Codensity and the ultrafilter monad by Tom Leinster. There he shows that the ultrafilter monad on the category $\mathrm{Set}$ ...

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222 views

### Spectral sequence from resolution of condensed abelian groups

I am watching Scholze's and Clausen's masterclass on Condensed Mathematics and I don't understand or can find any references on something they said.
You have a resolution
$$ \dots \to \mathbb{Z}[\...

**5**

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**1**answer

674 views

### Computations in condensed mathematics II, page 43

This is a follow up on my previous question for Lectures on Condensed Mathematics. I am reading ahead at page 43. But it is not directly clear to me from the results that:
How do we know $\Bbb Z[[T]],...

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**1**answer

954 views

### Computations in condensed mathematics, page 32-34

I started reading the Lectures on Condensed Mathematics. I am looking at the material at page 32-34. I have three fundamental computation questions:
At the last line of pg 32 - it seems to imply that ...

**16**

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704 views

### Why strong limit cardinals in the definition of condensed sets?

A condensed set à la Clausen–Scholze is, as far as I understand it, a small sheaf on the large site of profinite spaces. In Scholze's notes they are described as being objects of a category that is a ...

**5**

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209 views

### Is the filtered colimit topology on the space of signed Radon measures linear and locally convex?

Let $X$ be a compact Hausdorff space. In chapter 3 of Peter Scholze's Lectures on Analytic Geometry he considers the space of signed Radon measures on $X$ equipped with the filtered colimit (aka ...

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### 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 ...

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### Are there (enough) injectives in condensed abelian groups?

The question is very simple : does $Cond(\mathbf{Ab})$, the category of condensed abelian groups (as defined in Scholze's Lectures in Condensed Mathematics), have enough injectives ?
Does it, in fact,...

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667 views

### CG spaces from the perspective of sheaves over compact Hausdorff spaces

A compactly generated space is a space $X$ such that $f : X \rightarrow Y$ is continuous if and only if $K \rightarrow X \stackrel{f}{\rightarrow} Y$ is continuous for each compact hausdorff space $K$....