Ignoring set-theoretic problems, we can see condensed sets as sheaves of compact Hausdorff spaces. Using Gelfand Duality we obtain an equivalence of categories \begin{align*} \mathrm{CHaus}^{\mathrm{op}} \cong C^{\ast}\mathrm{comm}_1 \end{align*} where $C^{\ast}\mathrm{comm}_1$ is the category of commutative unital $C^{\ast}$ algebras. This implies that we can regard condensed sets as product-preserving functors $X: C^{\ast}\mathrm{comm}_1 \rightarrow \mathrm{Set}$ that satisfy the 'gluing axiom'.

$\textbf{Question 0:}$ Is it correct terminology to say that condensed sets are cosheaves of $C^{\ast}\mathrm{comm}_1$?

$\textbf{Question 1:}$ Does the gluing axiom translate to: Given a monomorphism $A \rightarrow B$ of $C^{\ast}\mathrm{comm}_1$, let $i_0, i_1$ be the two canonical maps $B \rightarrow B \otimes_A B$. Then, there is a bijection \begin{align*} X(A) \cong \{f \in X(B) \mid X(i_0) (f) = X(i_1) (f) \} \end{align*}

There is this general idea that the theory of $C^{\ast}$-algebras is noncommutative topology/geometry, with condensed sets we can take this idea quite literally and call noncommutative condensed sets to be a product-preserving functors from the category of unital $C^{\ast}$-algebras satisfying the gluing axiom.

$\textbf{Question 2: (Main Question)}$ Is this informal definition of noncommutative condensed set related to something in Noncommutative Geometry? What properties a definition of noncommutative space must have? Is there something interesting related to this in general?

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    $\begingroup$ Here is a basic question: whatever the category of "noncommutative condensed sets" is, the category of usual condensed sets (maybe with some niceness condition) should embed into it. Is every condensed set a restriction of a "noncommutative condensed set" in your sense to $C^*comm_1$? $\endgroup$
    – Wojowu
    Mar 20 at 2:39
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    $\begingroup$ Q0: yes, Q1: yes. I disagree however with the premise of Q2. From what I understand, much of Connes' motivation is to do geometry with "wild" quotient spaces. But many of the "wild" quotient spaces he cared about make sense as condensed sets (or maybe condensed groupoids). Also note that the gluing axiom does not even translate to the noncommutative case, as you can't form a $C^*$-algebra $B\otimes_A B$. $\endgroup$ Mar 20 at 13:37
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    $\begingroup$ Perhaps not quite what you're looking for, but we've explored such toposes over C*-algebras in our Bohr topos line of work ncatlab.org/nlab/show/Bohr+topos $\endgroup$ Mar 20 at 14:56
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    $\begingroup$ Thank you for the comments! Prof Scholze, rather than the tensor product, we could look at the pushout in the category of unital C*Algebra, which is the amalgamated free product. Prof Spitters, I was not aware of this, it looks very interesting. Thank you! $\endgroup$ Mar 20 at 15:40
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    $\begingroup$ @BasSpitters Perhaps not quite fit to this question, but do we have any sort of "noncommutative boolean algebras" which could replace $C^*$-algebras in question? $\endgroup$
    – Z. M
    Mar 20 at 17:57


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