Weak topologies on toposes (Q2657885)

Let \(\mathcal{E}\) be a topos and \(\Omega\) its subobject classifier. A \textit{weak topology} on \(\mathcal{E}\) is defined as a morphism \(j\colon \Omega\to\Omega\), such that 1) \(j\ \circ\) true = true, 2) \(j\circ\wedge\leq\wedge\circ(j\times j)\), in which \(\wedge\colon \Omega\times\Omega\to \Omega\) is the conjunction map and \(\leq\) stands for the internal order on \(\Omega\) that comes from the equalizer of \(\wedge\) and the first projection on \(\Omega\). If \(j\circ\wedge=\wedge\circ(j\times j)\), then a weak topology is called \textit{productive}. For a productive weak tpology \(j\) on \(\mathcal{E}\), the full subcategory \textbf{Sh}\(_j\mathcal{E}\) of all \(j\)-sheaves of \(\mathcal{E}\) is a topos. On a (co)complete topos, weak topologies form a complete residuated lattice. Weak topologies on a topos \(\mathcal{E}\) are one-to-one correspondence with modal closure operators defined on Sub\(_{\mathcal{E}}(E)\), the subobjects of \(E\), for all \(E\in \mathcal{E}\). An object \(C\) is called \(j\)-\textit{separated} if in every commutative diagram \(B\xrightarrow{m}A \overset{g}{\underset{g'}\rightrightarrows} C\), where \(m\colon B\to A\) is a \(j\)-dense monomorphism, we have \(g=g'\). For a productive weak topology, a left adjoint to the inclusion functor from the category \textbf{Sep}\(_j\mathcal{E}\) of all separated objects of \(\mathcal{E}\) to the full subcategory of \(\mathcal{E}\) consisting of all objects \(E\) of \(\mathcal{E}\) for which the closure of diagonal subobject \(\Delta_E\) of \(E\times E\) is closed, is established. The sheaf associated with any separated object with respect to a given productive weak topology is obtained. These concepts are applied to the topos \(M\)-\textbf{Sets}, \(M\) a monoid, and the subobject classifier \(\Omega_M\) consisting of the set of all right ideals \(K\) of \(M\). Then every left ideal \(I\) of \(M\) determines a weak topology \(j^I\colon \Omega_M\to\Omega_M\) given by \(j^I(K)=\{m\in M\vert\ \forall n\in I, mn\in K\}\).