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

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Mahmoudi Mojgan; Nejah Amir Homayoun

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Journal Paper

Paper Information

Journal: CATEGORIES AND GENERAL ALGEBRAIC STRUCTURES WITH APPLICATIONS Year:621 | Volume: | Issue: Page(s): 233-260

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Information Journal Paper

Title

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

Author(s)

Mahmoudi Mojgan | Nejah Amir Homayoun | Issue Writer Certificate

Keywords

Topological spaceQ4

frameQ4

$M$-setQ4

toposQ4

sober spaceQ4

spatial frameQ4

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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