Background. Let $X$ be an affine variety over a field $k$ of characteristic $0$. We can define then the loop space of $X$, denoted $\mathcal{L}X$, to be the functor taking a $k$-algebra $A$ to the set $X(A((t)))$. This is representable by an ind-scheme (of ind- infinite type). Intuitively this is the mapping space $\operatorname{Map}(D^{*},X)$, where $D^{*}$ is the formal punctured $k$-disk, $\operatorname{spec}k((t))$. This is not quite correct however, indeed for it to be correct we would need $\operatorname{spec}(A)\times D^{*}$ to be $\operatorname{spec}(A((t)))$, which it is not as soon as $A$ is infinite dimensional as a $k$ vector space. Nonetheless viewing it as a mapping space seems reasonable, in particular there is a well defined evaluation map from the universal correspondence. Note that for this to work we cannot naively take $\mathcal{L}X\times D^{*}$, again we have to complete the product. For example if $X=\mathbb{A}^{1}$ with coordinate $x$, then $\mathcal{O}(\mathcal{L}X\times D^{*})$ must contain the two way infinite Laurent series $x(t):=\sum_{i}x_{i}t^{i}$, where the $x_{i}$ are the natural coordinates on loop space. Notice that the Laurent tails tend to $0$ in $\mathcal{O}(\mathcal{L}X)$.

Question. Is there a natural algebro-geometric context in which $X$, $D^{*}$ and $\mathcal{L}X$ all live, so that we genuinely have $\operatorname{Map}(D^{*},X)\cong\mathcal{L}X$? Such a context must handle pro-discrete objects and adic ones in which a uniformiser is inverted. In particular can I use solid $k$-modules to build an appropriate context?

Naive guess. Thanks to Clausen-Scholze, we have a very well behaved symmetric monoidal category of solid $k$-modules. I believe that all the objects above sit naturally in this category, and that the products are as required. For example, if $A$ is a pro-discrete $k$-algebra, then $A\otimes_{\operatorname{solid}}k((t))$ is (as a module at least) the algebra of two way infinite Laurent series whose Laurent tails tend to $0$, according to Solid tensor product of pro-discrete space with Laurent series.

I cannot locate in the Clausen-Scholze work a definition of space modeled on the symmetric monoidal category $\operatorname{Solid}_{k}$, which leads me to believe that either it is a very special case of the analytic rings constructions, or that it is not useful.

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    $\begingroup$ They are adic spaces. The punctured open disk is discussed in §4.1 of Berkeley lecture notes. As for relation to condensed mathematics, see the chapter on discrete Huber pairs of Condensed.pdf and Analytic.pdf. $\endgroup$
    – Z. M
    Mar 27, 2022 at 9:18
  • $\begingroup$ @ZM ok so my understanding is that in the adic-spaces we do not have all pro-discrete algebras as "affine objects" but that if i consider Loops as an ind- adic thing then this is an honest mapping space etc? I was hoping a version of alg geom built from solid things would allow all three to be seen as affine $\endgroup$
    – EBz
    Mar 27, 2022 at 12:27
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    $\begingroup$ As mentioned in the other question, the solid tensor products do not quite work out this way. I haven't tried to look at loop groups from this perspective. $\endgroup$ Apr 8, 2022 at 21:39


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