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  "body": "\n## Definition\n\nA *profinite set* is a [[Stone space|ZV38HA5PAQ]], that is, a compact Hausdorff totally disconnected space.\nFix 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.\nA *$\\kappa$-condensed set* is a [[presheaf|AND2WHZC8P]] $T \\colon \\mathbf{ProFin}_\\kappa^{\\mathrm{op}} \\to \\mathbf{Set}$ such that:\n\n1. $T(\\emptyset)$ is a one-point set.\n2. 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.\n3. For each continuous surjection $p \\colon S' \\to S$, the map $T(p)$ is a bijection from $T(S)$ onto the [[equalizer|CTJ5AD8ST5]] of the two maps $T(S') \\rightrightarrows T(S' \\times_S S')$ induced by the projections.\n\nA *condensed set* is a $\\kappa$-condensed set for some such $\\kappa$.\nFormally, the category of condensed sets is the union (colimit) over $\\kappa$ of the categories of $\\kappa$-condensed sets.\n\n## Remarks\n\nThe three conditions say exactly that $T$ is a [[sheaf|RT8A8FJ566]] for the Grothendieck topology whose covering families are the finite jointly surjective families.\nThe cardinal $\\kappa$ only avoids set-theoretic size problems; other authors use sheaves on all compact Hausdorff spaces with a fixed universe.\nExample: for a topological space $X$, the functor $\\underline{X}(S) = C(S, X)$ of continuous maps is a condensed set.\nA continuous map on $S_1 \\sqcup S_2$ is a pair of continuous maps.\nA continuous map on $S'$ that agrees on the two pullbacks descends to $S$, because a surjection of compact Hausdorff spaces is a quotient map.\nNon-example: for a set $A$ with at least two points, the constant presheaf $T(S) = A$ fails clause 1, since $T(\\emptyset) = A$.\nThe value $T(*)$ at a point is the underlying set of the condensed set.\n",
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