$\newcommand\LRS{\mathsf{LRS}}\newcommand\FormalSch{\mathsf{FormalSch}}\DeclareMathOperator\Spf{Spf}\newcommand\IndSch{\mathsf{IndSch}}\newcommand\ALRS{\mathsf{ALRS}}\newcommand\FSch{\mathsf{FSch}}$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:Aformal schemeis 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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