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What is a noncommutative topos
In 1702.04949 noncommutative frames were introduced, generalizing the usual notion of frames of open sets of a topological space. In this paper we extend this notion to noncommutative GrothendieckExpand
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An arithmetic topos for integer matrices
Abstract 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 topos-theoretic points are given by the doubleExpand
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Azumaya representation schemes
TLDR
We extend Grothendieck topologies on commutative algebras to the category of all Azumaya algebnas and we show that the functor assigning to an Azumaaya algebra $A$ the set of all algebra maps $R \to A$ from a fixed $\mathbb{C}$-algebra $R$, is a sheaf. Expand
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A Topological Groupoid Representing the Topos of Presheaves on a Monoid
  • Jens Hemelaer
  • Mathematics, Computer Science
  • Appl. Categorical Struct.
  • 6 June 2019
TLDR
We give an alternative, more algebraic construction in the special case of a topos of presheaves on an arbitrary monoid. Expand
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Azumaya toposes
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 theExpand
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Degeneration in positive characteristic
2.1 Step A: Everything lifts globally . . . . . . . . . . . . . . . . . . 5 2.2 Step B: The relative Frobenius lifts only locally; decomposability in degrees 0 and 1 . . . . . . . . . . . . . . . . .Expand
Grothendieck topologies on posets
Lindenhovius has studied Grothendieck topologies on posets and has given a complete classification in the case that the poset is Artinian. We extend his approach to more general posets, byExpand
Azumaya geometry and representation stacks
We develop Azumaya geometry, which is an extension of classical affine geometry to the world of Azumaya algebras, and package the information contained in all quotient stacks $[\mathrm{rep}_nExpand
An essential, hyperconnected, local geometric morphism that is not locally connected.
TLDR
We give an example of an essential, hyperconnected, local geometric morphism that is not locally connected, arising from our work-in-progress on geometric morphisms $\mathbf{PSh}(M) <- PSh(N)$, where $M$ and $N$ are monoids. Expand
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