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arXiv · 2203.09133

Geometric morphisms between toposes of monoid actions: factorization systems

Jens Hemelaer·Morgan Rogers
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Abstract

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) $\rightarrow$ PSh(N) and properties of the semigroup homomorphisms M $\rightarrow$ 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, {\'e}tale morphisms, pure morphisms and complete spreads. We end with an application to topos-theoretic Galois theory to the special case of toposes of the form PSh(M).

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Jens Hemelaer, Morgan Rogers. 2022-03-17. Geometric morphisms between toposes of monoid actions: factorization systems. https://arxiv.org/abs/2203.09133

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