Topological spaces versus frames in the topos of $M$-sets

Document Type : Research Paper

Authors

1 Mojgan Mahmoudi; Department of Mathematics, Faculty of Mathematical Sciences, Shahid Beheshti University, Tehran 19839, Iran

2 Department of Mathematics, Faculty of Mathematical Sciences, Shahid Beheshti University, Tehran 19839, Iran

10.48308/cgasa.2023.234111.1455

Abstract

In this paper we study topological spaces, frames, and their confrontation in the presheaf topos of $M$-sets for a monoid $M$. We introduce the internalization, of the frame of open subsets for topologies, and
of topologies of points for frames, in our universe.
Then we find functors between the categories of topological spaces and of frames in our universe.
We show that, in contrast to the classical case, the obtained functors do not have an adjoint relation for a general monoid, but in some cases such as when $M$ is a group, they form an adjunction.
Furthermore, we define and study soberity and spatialness for our topological spaces and frames, respectively. It is shown that if $M$ is a group then the restriction of the adjunction to sober spaces and spatial frames becomes into an isomorphism.

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References

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Categories and General Algebraic Structures with Applications
Volume 20, Issue 1
Special Issue Dedicated to Prof. Themba Dube
January 2024
Pages 233-260

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