A guide to topological reconstruction on endomorphism monoids and polymorphism clones

TL;DR

This paper surveys recent progress in understanding when the algebraic structure of endomorphism monoids and polymorphism clones determines their topology, extending known results and clarifying the current state of this emerging research area.

Contribution

It provides a comprehensive overview of topological reconstruction results for endomorphism monoids and polymorphism clones, including new extensions of existing theorems.

Findings

Extended existing results on topological reconstruction

Clarified conditions under which algebraic structure determines topology

Surveyed recent developments in the field

Abstract

Various spaces of symmetries of a structure are naturally endowed with both an algebraic and a topological structure. For example, the automorphism group of a structure is, on top of being a group, a topological group when equipped with the topology of pointwise convergence. In some cases, the algebraic structure of such space alone is sufficiently rich to determine its topology (under some requirements on the topology). For automorphism groups, the problem of when this happens has been actively pursued over the last 40 years. With the exception of some early work of Lascar, the analogue of this problem for endomorphism monoids and polymorphism clones has only received attention in the past 15 years. In this guide, we survey the current state of affairs in this relatively young line of research. We moreover use this opportunity to polish several existing results and to extend them beyond what was hitherto known.