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 E, B an object of E, and j B = j × 1 B the induced topology on the slice topos E/B. In this manuscript, we analyze some properties of the pullback functor Π B : E → E/B which have deal with topology. Then for a left cancelable class M of all j-dense monomorphisms in a topos E, we achieve some necessary and sufficient conditions for that (M, M ⊥ ) is a factorization system in E, which is related to the factorization systems in slice topoi E/B, where B ranges over the class of objects of E. Among other things, we prove that an arrow f : X → B in E is a j B -sheaf whenever the graph of f , is a section in E/B as well as the object of sections S(f ) of f , is a j-sheaf in E. Furthermore, we introduce a class of monomorphisms in 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.