Internal parameterization of hyperconnected quotients (Q6595527)

One of the most fundamental facts in topos theory is the internal parameterization of subtoposes: the bijective correspondence between subtoposes of a given topos \(\mathcal{E}\) and Lawvere-Tierney topologies \(j:\Omega_{\mathcal{E}} \to \Omega_{\mathcal{E}},\) where \(\Omega_{\mathcal{E}}\) is its subobject classifier.\N\NIn this nice and well-written paper, the author tackles to an open problem proposed by F. W. Lawvere, which questions weather the quotients of a Grothendieck topos can be parametrized internally, similar to subtoposes. To do so, he defines a \textit{local state classifier} \(\Xi\) of a category \(\mathcal{C}\) as a colimit of the inclusion functor \({\mathcal{C}}_{\text{mono}}\to\mathcal{C}\), if it exists, where \({\mathcal{C}}_{\text{mono}}\) denote the subcategory of \(\mathcal{C}\) that consists of all monomorphisms and the same objects of \(\mathcal{C}\). The associated cocone is referred to as \(\{\xi_{X}: X\to \Xi\}_{X\in\operatorname{ob}({\mathcal{C}})}\).\N\NThe main theorem (Theorem 4.1) discussed in the paper is the internal parameterization of hyperconnected quotients (A geometric morphism \(f: \mathcal{E}\to\mathcal{F}\) is called hyperconnected if its inverse image functor \(f^*\) is fully faithful. This means that \(f^*\) preserves and reflects all morphisms between objects in the topos \(\mathcal{F}\)). Here, a hyperconnected geometric morphism from a topos \(\mathcal{E}\) is referred to as a hyperconnected quotient of \(\mathcal{E}\), emphasizing the aspect as a quotient of the topos \(\mathcal{E}\). This theorem establishes a bijective correspondence between hyperconnected quotient of \(\mathcal{E}\), internal filters of the local state classifier of \(\mathcal{E}\), and internal semilattice homomorphisms \(\Xi\to \Omega_{\mathcal{E}}\) (the subobject classfier of the topos \(\mathcal{E}\)). As a corollary of this theorem, the author give a partial solution to Lawvere's open problem; For a locally small Boolean topos with a local state classifier (e.g., an arbitrary Boolean Grothendieck topos), there exists an internal parameterization of quotients. In particular, the number of quotients is small.\N\NThe paper provides examples from graph theory, group actions, and sheaves over topological spaces to illustrate the concept of local state classifiers. It also addresses the classification of hyperconnected quotients in particular toposes like the topos of directed graphs and the topos of combinatorial species.\N\NThe paper concludes with two Appendices providing additional information about internal semilattices, filters, and the existence theorem for a local state classifier.