The least subtopos containing the discrete skeleton of \(\Omega\) (Q6593823)

This paper is concerned with the dimension theory outlined in [\textit{F. W. Lawvere}, Lect. Notes Math. 1488, 1--13 (1991; Zbl 0779.18001)]. This paper proves that, for a pre-cohesive geometric morphism \(p:\mathcal{E}\rightarrow\mathcal{S}\), the weak Aufhebung of level 0 exists and, moreover, it coincides with the least subtopos containing the 0-skeleton of \(\Omega\), which is analogous to the fact that, if \(\mathcal{S}\)\ is Boolean, then level 0 is the Aufhebung of level \(-\infty\), as already observed in [\textit{F. W. Lawvere} and \textit{M. Menni}, Theory Appl. Categ. 30, 909--932 (2015; Zbl 1375.18017), Corollary 4.5].