A topological proof that compact Hausdorff spaces are not finitely co-concrete

Abstract

Lieberman, Rosický, and Vasey proved that ${\mathbf{CompHaus}}^{\mathrm{op}}$ - the opposite of the category of compact Hausdorff spaces - is not finitely concrete by a route through Hilbert and Banach spaces, commutative unital $C^{\ast }$-algebras, and Gelfand duality. We give a short topological proof. Moreover, we strengthen the result by identifying a specific sequential colimit in ${\mathbf{CompHaus}}^{\mathrm{op}}$ that no faithful set-valued functor preserves.