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A Topological Groupoid Representing the Topos of Presheaves on a Monoid

Abstract

Butz and Moerdijk famously showed that every (Grothendieck) topos with enough points is equivalent to the category of sheaves on some topological groupoid. We give an alternative, more algebraic construction in the special case of a topos of presheaves on an arbitrary monoid. If the monoid is embeddable in a group, the resulting topological groupoid is the action groupoid for a discrete group acting on a topological space. For these monoids, we show how to compute the points of the associated topos.

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Acknowledgements

I would like to thank Karin Cvetko-Vah and Lieven Le Bruyn for the interesting discussions regarding the interpretation of étale spaces (over an Alexandrov-discrete space) in terms of posets. Further, I would like to thank the anonymous reviewer for their helpful comments.

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Correspondence to Jens Hemelaer.

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Communicated by Matías Menni.

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The author is a Ph.D. fellow of the Research Foundation—Flanders (FWO).

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Hemelaer, J. A Topological Groupoid Representing the Topos of Presheaves on a Monoid. Appl Categor Struct 28, 749–772 (2020). https://doi.org/10.1007/s10485-020-09596-9

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  • DOI: https://doi.org/10.1007/s10485-020-09596-9

Keywords

  • Topos theory
  • Butz–Moerdijk
  • Monoid
  • Topological groupoid
  • Topos points
  • F\(_1\)