$\newcommand\LRS{\mathsf{LRS}}\newcommand\FormalSch{\mathsf{FormalSch}}\DeclareMathOperator\Spf{Spf}\newcommand\IndSch{\mathsf{IndSch}}\newcommand\ALRS{\mathsf{ALRS}}\newcommand\FSch{\mathsf{FSch}}$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)?
Best Answer
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.