55
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What actually is the idea behind the condensed mathematics?
I don't pretend to have anything more than a superficial understanding of condensed mathematics, but Scholze's lecture notes (on condensed mathematics and analytic geometry) are so clearly written ...
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37
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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?
Good question!
We've been trying to figure this out as we went along, but so far unsuccessfully. Some more precise points:
For many (but definitely not all) applications to geometry over the real ...
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31
votes
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Examples of solid abelian groups
Here's a rule of thumb: As long as the construction is nonarchimedean and does not involve noncompleted tensor products, it's solid.
More precisely, anything you can build from discrete abelian groups ...
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30
votes
Nonconvexity and discretization
I was asked to answer this question. The people one really needs to ask here are the people in high-dimensional convex geometry/probability/statistics/computer science and metric geometry, but in any ...
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29
votes
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Condensed criterion for sheafiness of adic spaces
Thanks for the question! One interpretation of the conjecture is true. Let me elaborate. The following results are kind of implicit in some discussion towards the end of www.math.uni-bonn.de/people/...
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29
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Derived categories and $\infty$-categories necessary for condensed mathematics
There are several questions (implicit) here.
In the texts as they are written, how much knowledge on derived categories (as triangulated categories, or as stable $\infty$-categories) is assumed?
...
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28
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What is the precise relationship between pyknoticity and cohesiveness?
The work on analytic geometry is all joint with Dustin Clausen!
Your main question seems a little vague to me, but let me try to get at it by answering the subquestions. See also the discussion at ...
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27
votes
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Breen-Deligne packages and the liquid tensor experiment
The comments have already given the answers, but let me assemble them here with my account of the story.
When Scholze first posted the Liquid Tensor Experiment, it was quickly identified (by both ...
- 5,484
23
votes
Nonconvexity and discretization
I will start with a reformulation of the Proposition in the question in terms that are easier to digest for analysts.
Proposition: let $0<\tilde{r}<r<1$ and $0<p<1$ be such that $\tilde{...
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22
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On the connections between condensed mathematics and homotopy theory
The way in which "condensed sets are similar to topological spaces" is very different from the way in which "$\infty$-groupoids are similar to topological spaces". In fact, ...
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21
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Are there (enough) injectives in condensed abelian groups?
Indeed, there are no nonzero injective condensed abelian groups.
Let $I$ be an injective condensed abelian group. We can find some surjection
$$ \bigoplus_{j\in J} \mathbb Z[S_j]\to I$$
for some index ...
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21
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Condensed vs pyknotic vs consequential
Some comments:
Regarding 1): They are quite different. Johnstone actually uses a very general notion of "cover" in his sequential topos -- his site is a full subcategory of metrizable ...
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21
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Reference request for condensed math
Dagur Ásgeirsson has written a text to fill this gap:
We discuss in some detail the prerequisites for each of the first four chapters of Scholze's "Lectures on Condensed Mathematics". Some ...
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What intuitive notion is formalized by condensed mathematics?
There is an implicit notion in this question that condensed mathematics was specifically constructed to formalise some intuition about how to work with 'topology-like' structures. This is historically ...
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21
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What intuitive notion is formalized by condensed mathematics?
Condensed sets axiomatize the notion of convergence rather than the notion of neighborhoods. Unlike topological spaces, they allow a sequence to converge "for multiple different reasons". ...
- 1,838
19
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Properties of pyknotic sets
Let me recall a little bit of the background. The question is about the relation between topological spaces and pyknotic sets, and properties of the topos of pyknotic sets. Recall that pyknotic sets ...
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18
votes
What are the points (and generalized points) of the topos of condensed sets?
The category $\mathbf{Cond}$ of condensed sets is equivalent to the category of small sheaves over any of the following three large sites. (For small sheaves, see Mike Shulman's paper Exact ...
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16
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What does the topos of (light) condensed sets classify?
The topos of light condensed sets is generated by the Cantor set $\Delta = \prod_{\mathbb N} \{0,1\}$. So it classifies "Cantor space objects". Here is what this gives, essentially ...
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14
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Condensed / pyknotic approach to orbifolds?
Great question! I think the answer ought to be yes. But I must also make a disclaimer that I don't really know what orbifolds are, and the comments by David Roberts make be believe that what I thought ...
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13
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Computations in condensed mathematics, page 32-34
Correct, as both sides are the $S$-indexed direct sum of copies of $\mathbb{Z}$. For the LHS this holds by the universal property of $\mathbb{Z}[S]$, and for the RHS note that $C(S,\mathbb{Z}) = \...
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Witt vectors, the cotangent complex, and a solid construction of $B_{dR}^+$
$\newcommand\BdR{B_\text{dR}^+}\newcommand\Ainf{A_\text{inf}}$Thank you for the question! Actually, you've caught me out. Though I didn't realize it at the time, I was indeed cheating and should have ...
- 9,273
13
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Are condensed sets (locally) cartesian closed?
Condensed sets are indeed locally cartesian closed. On the other hand, for no cardinal $\kappa$ (no matter how inaccessible) the functor from $\kappa$-condensed sets to condensed sets preserves all ...
- 21.3k
12
votes
Infinity-categorical analogue of compact Hausdorff
That's a good question! I think Barwick and Haine have thought much more about this, and maybe they already know the answer? What I say below is definitely known to them. Also beware that I've written ...
- 21.3k
12
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Cohesion relative to a pyknotic/condensed base
Let me try to cut through the jargon. One thing that confuses me are two uses of "tangent spaces" here, that I believe are quite unrelated. One is the usual notion of tangent spaces of ...
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12
votes
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Condensed math and cofiltered limits
It should be noted that already in $\mathbf{Set}$, the free functor $\mathbf Z^{(-)} \colon \mathbf{Set} \to \mathbf{Ab}$ does not preserve cofiltered limits. For a cofiltered diagram $D \colon \...
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12
votes
Mixing solids and liquids
Good question!
I think the real context for the question was whether certain objects that are implicit in work of Darmon (and collaborators) could exist within this framework of analytic geometry. ...
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12
votes
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Is there a good theory of solid vector spaces?
I will prove that the result is true if $F$ is a finitely generated field, but fails if $F$ is countably generated field that is not finitely generated.
Let me first discuss the case $F=\mathbb Q$. ...
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11
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Reference request for condensed math
Blowing my own trumpet a bit: http://math.commelin.net/files/liquid_notes.pdf
These are notes for a talk that I gave this week for an audience of complex geometers.
I take a somewhat unorthodox ...
- 5,484
10
votes
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Different definitions of condensed sets
The question is not precise enough: it depends which topology you chose on the category of topological spaces. You will get the same category of sheaves if you are in a situation where Grothendieck's ...
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10
votes
Are absolute Galois groups condensed?
Like any profinite group (or much more general types of topological groups, such as compactly generated ones), you can consider it as a condensed group in the sense of condensed mathematics. In fact, ...
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