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?

2more comments