Questions tagged [condensed-mathematics]
Condensed mathematics of Clausen and Scholze. Closely related to the pyknotic mathematics of Barwick and Haine.
106
questions
3
votes
0
answers
333
views
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-...
5
votes
0
answers
359
views
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) $ ...
16
votes
0
answers
594
views
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 ...
8
votes
1
answer
1k
views
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 ...
3
votes
1
answer
356
views
"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 ...
0
votes
0
answers
211
views
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 ...
39
votes
1
answer
4k
views
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 ...
3
votes
0
answers
520
views
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?
11
votes
1
answer
665
views
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 ...
8
votes
1
answer
1k
views
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 ...
4
votes
0
answers
632
views
$\mathbb{Z}[T]$-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 ...
3
votes
0
answers
239
views
"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 ...
3
votes
1
answer
530
views
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 $(\:\underline{Y}\,)^X$ ...
5
votes
0
answers
221
views
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 ...
8
votes
0
answers
409
views
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{...