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 a 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.

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.

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 a 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 seeming to suggest otherwise.

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 a 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.