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

Abstract: 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 $\Pi_B:\mathcal{E}\rightarrow \mathcal{E}/B$ 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.