arXiv subjects

Zeinab Khanjanzadeh

Publications and source records attributed to Zeinab Khanjanzadeh.

3 recordsLinked to original sources

arXiv2020

Weak Topologies on Toposes

This paper deals with the notion of weak Lawvere-Tierney topology on a topos. Our motivation to study such a notion is based on the observation that the composition of two Lawvere-Tierney topologies is no longer idempotent, when seen as a closure operator. For a given topos $\mathcal{E}$, in this paper we investigate some properties of this notion. Among other things, it is shown that the set of all weak Lawvere-Tierney topologies on $\mathcal{E}$ constitutes a complete residuated lattice provided that $\mathcal{E}$ is (co)complete. Furthermore, when the weak Lawvere-Tierney topology on $\mathcal{E}$ preserves binary meets we give an explicit description of the (restricted) associated sheaf functor on $\mathcal{E}$.

math.CT↗

arXiv2017

Action preserving (weak) topologies on the category of presheaves

Let $\mathcal{C}$ be a finitely complete small category. In this paper, first we construct two weak (Lawvere-Tierney) topologies on the category of presheaves. One of them is established by means of a subfunctor of the Yoneda functor and the other one, is constructed by an admissible class on $\mathcal{C}$ and the internal existential quantifier in the presheaf topos $\widehat{\mathcal{C}}$. Moreover, by using an admissible class on $\mathcal{C},$ we are able to define an action on the subobject classifier $Ω$ of $\widehat{\mathcal{C}}$. Then we find some necessary conditions for that the two weak topologies and also the double negation topology $\neg\neg$ on $\widehat{\mathcal{C}}$ to be action preserving maps. Finally, among other things, we constitute an action preserving weak topology on $\widehat{\mathcal{C}}$.

math.CT↗

arXiv2016

Lawvere-Tierney sheaves, factorization systems, sections and $j$-essential monomorphisms in a topos

Let $j$ be a Lawvere-Tierney topology (a topology, for short) on an arbitrary topos $\mathcal{E}$, $B$ an object of $\mathcal{E}$, and $j_B = j\times 1_B$ the induced topology on the slice topos $\mathcal{E}/B$. In this manuscript, we analyze some properties of the pullback functor $Π_B:\mathcal{E}\rightarrow \mathcal{E}/B$ which have deal with topology. Then for a left cancelable class $\mathcal{M}$ of all $j$-dense monomorphisms in a topos $\mathcal{E}$, we achieve some necessary and sufficient conditions for that $(\mathcal{M} , \mathcal{M}^{\perp})$ is a factorization system in $\mathcal{E}$, which is related to the factorization systems in slice topoi $\mathcal{E}/B,$ where $B$ ranges over the class of objects of $\mathcal{E}$. Among other things, we prove that an arrow $f : X\rightarrow B$ in $\mathcal{E}$ is a $j_B$-sheaf whenever the graph of $f$, is a section in $\mathcal{E}/B$ as well as the object of sections $S(f)$ of $f$, is a $j$-sheaf in $\mathcal{E}$. Furthermore, we introduce a class of monomorphisms in $\mathcal{E}$, which we call them $j$-essential. Some equivalent forms of those and some of their properties are presented. Also, we prove that any presheaf in a presheaf topos has a maximal essential extension. Finally, some similarities and differences of the obtained result are discussed if we put a (productive) weak topology $j$, studied by some authors, instead of a topology.

math.CT↗