TY - JOUR
ID - 104105
TI - Topological spaces versus frames in the topos of $M$-sets
JO - Categories and General Algebraic Structures with Applications
JA - CGASA
LA - en
SN - 2345-5853
AU - Mahmoudi, Mojgan
AU - Nejah, Amir H.
AD - Mojgan Mahmoudi;
Department of Mathematics, Faculty of Mathematical Sciences,
Shahid Beheshti University, Tehran 19839, Iran
AD - Department of Mathematics, Faculty of Mathematical Sciences, Shahid Beheshti University, Tehran 19839, Iran
Y1 - 2024
PY - 2024
VL - 20
IS - 1
SP - 233
EP - 260
KW - Topological space
KW - frame
KW - $M$-set
KW - topos
KW - sober space
KW - spatial frame
DO - 10.48308/cgasa.2023.234111.1455
N2 - 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.
UR - https://cgasa.sbu.ac.ir/article_104105.html
L1 - https://cgasa.sbu.ac.ir/article_104105_8ef864d498af1086a8d29125553460a3.pdf
ER -