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

Marco Abbadini

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

Category Theory 2026-07-31 v1 General Topology

Authors: Marco Abbadini

Abstract

Lieberman, Rosick\'y, 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 $C^*$ -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.

Keywords

compact Hausdorff space topological proof finitely concrete $\mathbf{CompHaus}^{\mathrm{op}}$ Gelfand duality

Cite

@article{arxiv.2607.29103,
  title  = {A topological proof that compact Hausdorff spaces are not finitely co-concrete},
  author = {Marco Abbadini},
  journal= {arXiv preprint arXiv:2607.29103},
  year   = {2026}
}