GEOMETRIC MORPHISMS BETWEEN TOPOSES OF MONOID ACTIONS: FACTORIZATION SYSTEMS.

Let M, N be monoids, and PSh(M), PSh(N) their respective categories of right actions on sets. In this paper, we systematically investigate correspondences between properties of geometric morphisms PSh(M) → PSh(N) and properties of the semigroup homomorphisms M → N or flat-left-N-right-M-sets inducing them. More specifically, we consider properties of geometric morphisms featuring in factorization systems, namely: surjections, inclusions, localic morphisms, hyperconnected morphisms, terminal-connected morphisms, ´etale morphisms, pure morphisms and complete spreads. We end with an application of topos-theoretic Galois theory to the special case of toposes of the form PSh(M). [ABSTRACT FROM AUTHOR]