Different definitions of condensed sets

Question

The $\kappa$-condensed sets are defined as the sheaves on the site of profinite spaces of cardinality less than $\kappa$ (with $\kappa$ an uncountable strong limit cardinal) with morphisms the continuous maps, and whose covers are finite collections of jointly surjective maps.

I understand that you get the same category of condensed sets if you instead take the larger category of compact hausdorff spaces of cardinality less than $\kappa$, or the smaller category of Stone-Cech compactifications of sets of cardinality less than $\kappa$.

What happens if you use the "same site" on the category of all topological spaces of cardinality less than $\kappa$? Does this also yield the condensed sets? If not, what goes wrong?

Answer 1

The question is not precise enough: it depends which topology you chose on the category of topological spaces. You will get the same category of sheaves if you are in a situation where Grothendieck's comparison lemma applies.

That is, you need to chose a topology on the category of all topological spaces, that induces the correct topology when restricted to compact spaces, and such that for each topological space X, the set of all maps $K \rightarrow X$ with $K$ compact are a covering.

There definitely is a (actually unique) topology that satisfies this, and others that don't, and this is a necessary and sufficient conditions to get the equivalence of the categories of sheaves.

The unique topology for which this is going to work can be described as follow:

A collection of maps $X_i \to X$ is a covering (generates a covering sieve) if and only for each profinite space K and each continuous map $K \to X$, there is a finite covering $K_j \to K$ by compact spaces $(K_j)_{j \in J}$ such that each map $K_j \to X$ factors through one of the $X_i$.