A Topological Groupoid Representing the Topos of Presheaves on a Monoid
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A Topological Groupoid Representing the Topos of Presheaves on a Monoid Jens Hemelaer1 Received: 20 September 2019 / Accepted: 24 February 2020 © Springer Nature B.V. 2020
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. Keywords Topos theory · Butz–Moerdijk · Monoid · Topological groupoid · Topos points · F1
Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Equivariant Spaces Described by a Monoid . . . . . . . . . . 2.1 From Equivariant Sheaves to Monoid Actions . . . . . . 2.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Description in Terms of Posets . . . . . . . . . . . . . . 3 Converse Construction . . . . . . . . . . . . . . . . . . . . . 3.1 From Monoid Actions to Equivariant Sheaves . . . . . . 3.2 Explicit Translations . . . . . . . . . . . . . . . . . . . 3.3 Example: N-Sets . . . . . . . . . . . . . . . . . . . . . 4 Arbitrary Monoids . . . . . . . . . . . . . . . . . . . . . . . 4.1 Alexandrov Groupoids . . . . . . . . . . . . . . . . . . 4.2 Alexandrov Groupoid Associated to an Arbitrary Monoid 4.3 Proof of the Main Theorem . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Communicated by Matías Menni. The author is a Ph.D. fellow of the Research Foundation—Flanders (FWO).
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Jens Hemelaer [email protected] Department of Mathematics, University of Antwerp, Middelheimlaan 1, 2020 Antwerp, Belgium
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J. Hemelaer
1 Introduction In [1], Butz and Moerdijk showed that for every (Grothendieck) topos T with enough points, we can find a topological groupoid G such that T Sh(G),
where Sh(G) is the category of sheaves on G (also called the classifying topos of G). We give an alternative construction in the case that T = M-Sets for M a monoid. Here M-Sets is the topos with • as objects the sets S equipped with a left M-action; • as morphisms the functions f : S → S such that f (m · s) = m · f (s) for all m ∈ M
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