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.
We systematically investigate, for a monoid M, how topos-theoretic properties of $${{\,\mathrm{\mathbf {PSh}}\,}}(M)$$
PSh
(
M
)
, including the properties of being atomic, strongly compact, local, totally connected or cohesive, correspond to semigroup-theoretic properties of M.
We study the topos of sets equipped with an action of the monoid of regular 2×2 matrices over the integers. In particular, we show that the topostheoretic points are given by the double quotient GL 2 ( Z)\ M 2 (A f )/ GL 2 (Q), so they classify the groups Z 2 ⊆ A ⊆ Q 2 up to isomorphism. We determine the topos automorphisms and then point out the relation with Conway's big picture and the work of Connes and Consani on the Arithmetic Site. As an application to number theory, we show that classifying extensions of Q by Z up to isomorphism relates to Goormaghtigh conjecture.
In [4], many different Grothendieck topologies were introduced on the category of Azumaya algebras. Here we give a classification in terms of sets of supernatural numbers. Then we discuss the associated categories of sheaves and their topos-theoretic points, which are related to UHF-algebras. The sheaf toposes that correspond to a single supernatural number have an alternative description, involving actions of the associated projective general linear group.
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