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 B^∗ : \(\mathcal{E}\) \(\longrightarrow\) \(\mathcal{E}\)/B which are dealing with topologies.  Then for the left cancellable class \(\mathcal{M}\) of all j-dense monomorphisms in a topos \(\mathcal{E}\), we achieve some necessary and sufficient conditions for that the pair (\(\mathcal{M}\),\(\mathcal{M}\)^⊥) 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 \(\longrightarrow\) B in \(\mathcal{E}\) is a j_B-sheaf in \(\mathcal{E}\)/B 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 each member of the class j-essential.  Some equivalent forms 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.