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$\newcommand\LRS{\mathsf{LRS}}\newcommand\FormalSch{\mathsf{FormalSch}}\DeclareMathOperator\Spf{Spf}\newcommand\IndSch{\mathsf{IndSch}}\newcommand\ALRS{\mathsf{ALRS}}\newcommand\FSch{\mathsf{FSch}}$I'm trying to understand whether there's a fully faithful functor $\LRS \supset \FormalSch \to \IndSch$ and in what sense. Here's my progress so far:

Let $\mathsf{A}$ be the category of adic rings. The objects are topological rings whose topology is generated by a descending filtration of ideals whose intersection is $\{0\}$. Morphisms are continuous homomorphism of rings.

There's a functor $\Spf: \mathsf{A} \to \IndSch$ which takes an adic ring to the formal spectrum which is naturally a filtered colimit of (affine) schemes). The target of the functor could be that of adic locally ringed spaces (topological spaces with sheaves of adic rings and morphisms between for which the comorphism of sheaves is continuous). Denote this category $\ALRS$.

In $\ALRS$ we have an adjunction with the "continuous" global section functor $\Gamma_{\text{cont}} \dashv \Spf $. Continuous here just means it remembers the topology (i.e. the filtration).

Now the definition of formal schemes feels inevitable:

Definition: A formal scheme is an adic locally ringed space locally isomorphic to a formal spectrum of an adic ring. Denote the subcategory of formal schemes by $\FSch\subset \ALRS$.

This raises a problem though. There's no obvious way to turn a "formal scheme" in this sense into an ind-schemes (which are much more convenient for certain purposes). We could try to define the ind-scheme as the formal colimit over the Čech nerve of a chosen covering by formal spectra (which are themselves filtered colimits of affine schemes). However, this is probably a very bad idea since it will most likely depend on the choice of covering.

Question: Can we construct a functor $\FSch \to \IndSch$ with some good properties? (Hopefully fully faithful but if not maybe at least full.) If not is there a better definition of a formal scheme which enables you to play in both worlds (ind schemes and locally ringed spaces)?

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    $\begingroup$ I think without finiteness hypotheses you're doomed? There's a careful discussion of this stuff in the stacks project, where they discuss what conditions you need to have this nice Ind-presentation you're looking for. stacks.math.columbia.edu/tag/0AIT seems relevant, but I guess they don't focus much on morphisms $\endgroup$
    – Dylan Wilson
    Commented Apr 15, 2016 at 20:17
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    $\begingroup$ @SaalHardali Also, a ringed topos is a general type of answer for classical modulii problems (is there any other global way to speak about orbifolds?). And the geometric picture, having its merits, quickly gets messy with details. Personally I already have trouble visualizing formal schemes. Going further, if I consider a modulii problem for formal schemes, then I should represent it by what? Topoi with local adic rings? Is it really better than sheaves? $\endgroup$
    – Anton Fetisov
    Commented Apr 17, 2016 at 17:06
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    $\begingroup$ Would you mind undeleting this question? While no one has given an answer, I think it is an interesting question, and the voting indicates that other people agree. $\endgroup$
    – S. Carnahan ♦
    Commented Dec 29, 2016 at 1:40
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    $\begingroup$ Your category of adic rings seems "wrong" (what motivates your definition?). Use (pre)admissible rings as in EGA 0$_{\rm{I}}$, 7.2.1-7.2.2. Upon reading 10.5.1 and 10.6 in EGA I, the content of your question is this: does every formal scheme (in the sense of EGA) admit a fundamental system of ideals of definition? This is clearly affirmative in the locally noetherian case (see 10.6.3 and 10.6.10 in EGA I), and for the rather less evident qcqs case (as @DylanWilson suggested might work) see stacks.math.columbia.edu/tag/0AJF for an affirmative proof. $\endgroup$
    – nfdc23
    Commented Dec 17, 2017 at 13:31
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    $\begingroup$ The definition of (pre)admissible ring in EGA includes a condition involving powers of ideals whereas your proposed definition includes no such condition. You can use that distinction to make many (even noetherian) examples showing the definitions are not equivalent; good exercise. The notion you define is what EGA (and other places) call a "linearly topologized" ring (except that your notion is also Hausdorff); this is a weaker concept than what EGA defines as a (pre)"admissible" ring (which is the more appropriate notion to use). $\endgroup$
    – nfdc23
    Commented Dec 22, 2017 at 16:27

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In Yasuda's article Non-adic formal schemes, https://arxiv.org/abs/0711.0434, he redefines formal schemes as certain topological spaces equipped with a sheaf of pro-rings (as opposed to a sheaf of topological rings).

Complete linearly topologized rings embed fully faithfully into the category of pro-rings (Paragraph 5.2 of loc.cit) so maybe this is in the direction you are looking for? Admissibility is discussed in Paragraph 5.3.

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