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Geometric morphisms between toposes of monoid actions: factorization systems
Theory and applications of categories, Bunge Festschrift, Tome 40 (2024), pp. 80-129 Cet article a éte moissonné depuis la source Theory and Applications of Categories website

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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, étale 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).

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Classification : 18B25, 20M30
Keywords: topos, monoid, factorization, terminal-connected, étale, pure, complete spread 
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     author = {Jens Hemelaer and Morgan Rogers},
     title = {Geometric morphisms between toposes of monoid actions: factorization systems},
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Jens Hemelaer; Morgan Rogers. Geometric morphisms between toposes of monoid actions: factorization systems. Theory and applications of categories, Bunge Festschrift, Tome 40 (2024), pp. 80-129. http://geodesic.mathdoc.fr/item/TAC_2024_40_a3/