Planar Heyting Algebras for Children 2: Local Operators, J-Operators, and Slashings

TL;DR

This paper develops a visual and algebraic framework for understanding sheaf notions in topos theory using local operators, J-operators, slashings, and graph representations, extending previous toy models.

Contribution

It introduces a new representation of sheafness concepts via maps on Heyting Algebras, connecting local operators, J-operators, slashings, and graph models in topos theory.

Findings

Representations of sheafness as manageable maps J:H→H

Equivalence between slashings on Heyting Algebras and sets of question marks

Correspondence between two-column graphs and the logic of associated topoi

Abstract

Choose a topos $E$. There are several different "notions of sheafness" on $E$. How do we visualize them? Let's refer to the classifier object of $E$ as $\Omega$, and to its Heyting Algebra of truth-values, $Sub(1_E)$, as $H$; we will sometimes call $H$ the "logic" of the topos. There is a well-known way of representing notions of sheafness as morphisms $j:\Omega\to \Omega$, but these `$j$'s yield big diagrams when we draw them explicitly; here we will see a way to represent these `$j$'s as maps $J:H\to H$ in a way that is much more manageable. In the previous paper of this series we showed how certain toy models of Heyting Algebras, called "ZHAs", can be used to develop visual intuition for how Heyting Algebras and Intuitionistic Propositional Logic work; here we will extend that to sheaves. The full idea is this: notions of sheafness correspond to local operators and vice-versa; local operators correspond to J-operators and vice-versa; if our Heyting Algebra $H$ is a ZHA then J-operators correspond to slashings on $H$, and vice-versa; slashings on $H$ correspond to "sets of question marks" and vice-versa, and each set of question marks induces a notion of erasing and reconstructing, which induces a sheaf. Also, every ZHA $H$ corresponds to an (acyclic) 2-column graph, and vice-versa, and for any two-column graph $(P,A)$ the logic of the topos $\mathbf{Set}^{(P,A)}$ is exactly the ZHA $H$ associated to $(P,A)$.