First I apologize as the following is likely littered with misunderstanding.


My understanding at least from the discussion in Scholze-Weinstein is that inverse limits in the category of adic spaces rarely exist because there isn't a canonical way to define a topology on the direct limit of affinoid rings. Because of this, Scholze defines a weak notion of inverse limit which he denotes by $\sim$. If such a weak inverse limit $X$ is perfectoid, the situation is better as $X$ satisfies a universal property and thus is unique in the category of perfectoid spaces.


My question is do inverse limits always exist in the category of perfectoid spaces? My guess would be no because of the above. However, I am confused by language immediately before II.3.11 and before III.2.34 in Scholze's torsion paper seeming to suggest otherwise.



Inverse Limits in the category of Perfectoid Spaces
As mentioned by Scholze in [this lecture](https://www.msri.org/workshops/731/schedules/17596) (around 40:00; uniform Tate rings are called "spectral rings" in the lecture), given a directed system of maps of uniform Tate-Huber pairs $(R\_i,R\_i^+) \to (S,S^+)$, the map $\varinjlim\_i R\_i \to S$ is continuous for the $\varpi$-adic topology on $\varinjlim\_i R\_i^+$, where $\varpi$ is the image of a pseudouniformizer of some $R\_i$. Here we need to use the fact that $S$ is uniform, so $S^+$ is a ring of definition and it has the $\varpi$-adic topology.


So filtered inverse limits should exist in the category of (sous)perfectoid spaces since (sous)perfectoid rings are stably uniform. Unfortunately I do not know of a reference that explicitly proves this, but it does seem to be used implicitly in *[Etale cohomolgy of diamonds](http://www.math.uni-bonn.de/people/scholze/EtCohDiamonds.pdf)*.


