A condensed set à la Clausen–Scholze is, as far as I understand it, a small sheaf on the large site of profinite spaces. In [Scholze's notes](http://www.math.uni-bonn.de/people/scholze/Condensed.pdf) they are described as being objects of a category that is a large sequential colimit of toposes, each of which is $\mathrm{Sh}(\*\_{\kappa\text{-proét}})$, where the index $\kappa$ runs over the uncountable strong limit cardinals. The $\kappa$-proétale site $\*\_{\kappa\text{-proét}}$ consists of the profinite spaces whose underlying set is smaller than $\kappa$ (the topology is for now not important). The inclusions of sites induce fully faithful inverse image functors between the toposes, and the colimit defining the condensed sets is given by the colimit along these inverse image functors.


We can concretely describe this colimit as having objects pairs $(\kappa,F)$ consisting of an uncountable strong limit cardinal and a sheaf on $\*\_{\kappa\text{-proét}}$, where a map $(\kappa,F) \to (\kappa',G)$ is a map of sheaves on $\*\_{\lambda\text{-proét}}$ where $\lambda = \max\{\kappa,\kappa'\}$ where we have included the sheaf in the smaller topos into the larger topos (i.e. extended it to the larger site in the usual way).


Now I would like to know what goes wrong if I don't take this limit only over uncountable strong limit cardinals, but rather over all cardinals. (In this setup, a condensed set would be a pair consisting of an arbitrary cardinal $\mu$ and a sheaf on $\*\_{\mu\text{-proét}}$; morphisms are as above.) The strong limit cardinals just give a cofinal sequence, and it's not clear to me that the result is not equivalent. If there are specific properties of strong limit cardinals that are used, what are they?



Why strong limit cardinals in the definition of condensed sets?
Let me write down an answer to mark the question as answered.


As David points out, the definition would not change if one used all cardinals $\kappa$ instead of the strong limit ones, as the latter are cofinal in the former.


But strong limit cardinals are used to study the "individual layers" appearing in the definition : $\kappa$-condensed sets/abelian groups etc.


To prove that one can compare those to sheaves on $\kappa$-small profinite sets, or $\kappa$-small extremally disconnected sets, one uses that $\kappa$ is strong limit; see propositions 2.3, 2.5 of the notes and their proofs.


In turn, this comparison is very useful technically speaking, as sheaves on the site of $\kappa$-small extremally disconnected sets are just product preserving functors, and thus the category of sheaves enjoys very nice properties (with respect to co/limits for instance); and also the functors that connect these categories of sheaves are also particularly well-behaved, which allows one to make "local" computations of co/limits, which is again useful to prove specific things about their behaviour. For examples of this, see proposition 2.9 and theorem 2.2 of the notes.


As Peter pointed out in the comments, another thing that's good in $\kappa$-condensed abelian groups that doesn't work in general, but does if $\kappa$ is strong limit, is the fact that they are generated by compact projectives.


