Monoid Properties as Invariants of Toposes of Monoid Actions
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Monoid Properties as Invariants of Toposes of Monoid Actions Jens Hemelaer1
· Morgan Rogers2
Received: 14 May 2020 / Accepted: 13 November 2020 © The Author(s) 2020
Abstract We systematically investigate, for a monoid M, how topos-theoretic properties of PSh(M), including the properties of being atomic, strongly compact, local, totally connected or cohesive, correspond to semigroup-theoretic properties of M. Keywords Topos · Presheaves · Monoid · Flat · Projective · Semigroup · Local · Strongly compact · Colocal · Totally connected · Strongly connected · de Morgan · Cohesive
Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Toposes of Discrete Monoid Actions . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Tensors and Homs, Flatness and Projectivity . . . . . . . . . . . . . . . . . . . . . 1.2.1 Hom-Sets and Projectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Tensors and Flatness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Properties of the Global Sections Morphism . . . . . . . . . . . . . . . . . . . . . . . 2.1 Groups and Atomicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Right-Factorable Finite Generation and Strong Compactness . . . . . . . . . . . . 2.3 Right Absorbing Elements and Localness . . . . . . . . . . . . . . . . . . . . . . 2.4 The Right Ore Condition, Preservation of Monomorphisms and de Morgan Toposes 2.5 Spans and Strong Connectedness . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Preserving Exponentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Right Collapsibility and Total Connectedness . . . . . . . . . . . . . . . . . . . . 2.8 Left Absorbing Elements and Colocalness . . . . . . . . . . . . . . . . . . . . . . 2.9 Zero Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.10 Trivialising Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Communicated by Jirí Rosický.
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Morgan Rogers [email protected] Jens Hemelaer [email protected]
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Department of Mathematics, University of Antwerp, Middelheimlaan 1, 2020 Antwerp, Belgium
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Marie Sklodowska-Curie fellow of the Istituto Nazionale di Alta Matematica, Università degli Studi dell’Insubria, Via Valleggio n. 11, 22100 Como, CO, Italy
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J. Hemelaer, M. Rogers 3 Conclusion . . . . . . . . . . . . . . 3.1 Notable Omissions . . . . . . . 3.2 Relativisation and Generalisation References . . . . . . . . . . . . . . . .
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