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Geometric morphisms between toposes of monoid actions: factorization systems

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Jens Hemelaer and Morgan Rogers

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).

Keywords:
topos, monoid, factorization, terminal-connected, étale, pure, complete spread

2020 MSC:
18B25, 20M30

*Theory and Applications of Categories,*
Vol. 40, 2024,
No. 4, pp 80-129.

Published 2024-03-28.

http://www.tac.mta.ca/tac/volumes/40/4/40-04.pdf

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