Questions tagged [condensed-mathematics]
Condensed mathematics of Clausen and Scholze. Closely related to the pyknotic mathematics of Barwick and Haine.
31
questions
7
votes
0answers
263 views
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
votes
1answer
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 ...
2
votes
1answer
477 views
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 $...
3
votes
1answer
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 ...
0
votes
0answers
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
votes
1answer
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 ...
5
votes
1answer
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
votes
1answer
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
votes
1answer
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}^\...
10
votes
2answers
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\...
1
vote
1answer
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
votes
2answers
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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
votes
2answers
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
votes
1answer
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
votes
1answer
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
votes
1answer
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 ...
8
votes
1answer
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
votes
1answer
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 ...
1
vote
1answer
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,...
21
votes
1answer
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&...
3
votes
1answer
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
votes
1answer
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 ...
24
votes
1answer
1k views
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 $\mathrm{Set}$ ...
3
votes
0answers
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
votes
1answer
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]],...
10
votes
1answer
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
votes
1answer
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
votes
1answer
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 ...
23
votes
2answers
3k 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 ...
14
votes
1answer
2k views
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,...
7
votes
2answers
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 $K$....
