As is probably clear from my previous questions, I am coming to "condensed mathematics" from the naive perspective of a category theorist, without much knowledge of the intended applications in algebraic geometry and functional analysis. I hope therefore that this question is not *too* naive, but I haven't found a place yet where it is addressed clearly for a "lay" audience.

According to my current understanding, there are four categories that achieve something similar:

- The category of $\kappa$-condensed sets, for some cardinal $\kappa$, which are sheaves on the site of compact Hausdorff spaces of cardinality $<\kappa$.
- The category of pyknotic sets, which are the $\kappa$-condensed sets when $\kappa$ is an inaccessible.
- In the opposite direction (making the site smaller), Johnstone's topological topos, which is the category of sheaves on the full subcategory of spaces containing the point and the one-point compactification of $\mathbb{N}$.
- The category of condensed sets, which is the colimit of the categories of $\kappa$-condensed sets over all $\kappa$, or equivalently the category of "small sheaves" on the large site of all compact Hausdorff spaces.

I want to understand the differences between these categories, and why and in what situations one might choose one over the others. Specifically:

Johnstone's topological topos seems closely related to the category of $\aleph_1$-condensed sets. (The references I've seen restrict $\kappa$ to be a strong limit cardinal, but at least the definition seems to make sense for any cardinal.) It seems too much to hope for that they would be equivalent, but are they related by an adjunction at least? How "close" are they?

On that note, why is $\kappa$ usually restricted to be a strong limit cardinal?

To a pure category theorist, the first three categories have the obvious advantage over the fourth that they are toposes. In fact they are even

*local*toposes: their forgetful functor to sets has a right adjoint as well as a left adjoint. Why might one nevertheless choose to work with the non-local non-topos of condensed sets instead of one of these three toposes?Relatedly, for what applications is it

*not*sufficient to work with $\kappa$-condensed sets for a fixed small $\kappa$, like the smallest uncountable strong limit $\beth_\omega$? Or, for that matter, Johnstone's topological topos? In particular, are there desirable*abstract*properties that these categories lack? Or are there constructions that would give the "wrong" result when performed in these categories (and if so, in what sense)? Or is it that there are important examples of compactly generated spaces that aren't "$\kappa$-compactly generated" for small $\kappa$ and thus don't embed fully-faithfully in these categories?