Minimal and intrinsic topologies on monoids of elementary embeddings

Abstract

To every $\omega$-categorical structure $M$ one can associate two spaces of symmetries which determine the structure up to first-order bi-interpretability: the topological group $\mathrm{Aut}(M)$ of its automorphisms and the topological monoid $\mathrm{EEmb}(M)$ of its elementary embeddings, both equipped with the topology of pointwise convergence $\tau_{\mathrm{pw}}$. We investigate the relation of $\tau_{\mathrm{pw}}$ to other topologies on these spaces: in particular, when $\tau_{\mathrm{pw}}$ is minimal, i.e.~does not admit any strictly coarser Hausdorff semigroup topology. A common method to prove minimality of $\tau_{\mathrm{pw}}$ on $\mathrm{EEmb}(M)$ is to show that it coincides with the algebraically defined semigroup Zariski topology $\tau_{\mathrm{Z}}$. We show that $\tau_{\mathrm{pw}}$ differs from $\tau_{\mathrm{Z}}$ on $\mathrm{EEmb}(M)$ whenever $\mathrm{Aut}(M)$ has non-trivial centre. We then provide general conditions on the behaviour of algebraic closure on $M$ that imply minimality of $\tau_{\mathrm{pw}}$. These condition cover, for example, countable vector spaces and projective spaces over finite fields. Turning to $\mathrm{Aut}(M)$, we describe the minimal $T_1$ semigroup topologies on the automorphism groups of model-theoretically simple one-based $\omega$-categorical structures with weak elimination of imaginaries. We conclude by proving that the metric pointwise topology $\tau_{\mathrm{mpw}}$ is minimal, equals $\tau_{\mathrm{Z}}$, and is strictly coarser than $\tau_{\mathrm{pw}}$, on $\mathrm{EEmb}(M)$ for the real and the rational Urysohn space and sphere.