Tootfinder

Minimal and intrinsic topologies on monoids of elementary embeddings
J. de la Nuez Gonzalez, Zaniar Ghadernezhad, Paolo Marimon, Michael Pinsker
https://arxiv.org/abs/2603.28419 https://arxiv.org/pdf/2603.28419 https://arxiv.org/html/2603.28419
arXiv:2603.28419v1 Announce Type: new
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.
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Scholarly (history) tables of contents, EN vs FR:
English: no more than 3 words per chapter title. These are merely signifiers, empty, regardless of length, until one has digested the argument each lays out. Why waste ink and paper on such pointless abstraction?
French: describing a chapter in fewer words than are contained in the chapter betrays the essence of the work. Still, we know that nothing is perfect and brevity has its merits so here is our 27-page table of contents.

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