Condensed set

Definition

A profinite set is a Stone space, that is, a compact Hausdorff totally disconnected space. Fix an uncountable strong limit cardinal $\kappa$, and let $\mathbf{ProFin}_\kappa$ be the category of profinite sets of cardinality less than $\kappa$, with continuous maps. A $\kappa$-condensed set is a presheaf $T \colon \mathbf{ProFin}_\kappa^{\mathrm{op}} \to \mathbf{Set}$ such that:

1. $T(\emptyset)$ is a one-point set.
2. For all $S_1, S_2$, the map $T(S_1 \sqcup S_2) \to T(S_1) \times T(S_2)$ is a bijection.
3. For each continuous surjection $p \colon S' \to S$, the map $T(p)$ is a bijection from $T(S)$ onto the equalizer of the two maps $T(S') \rightrightarrows T(S' \times_S S')$ induced by the projections.

A condensed set is a $\kappa$-condensed set for some such $\kappa$. Formally, the category of condensed sets is the union (colimit) over $\kappa$ of the categories of $\kappa$-condensed sets.