Each closure operator induces a topology and vice-versa ("version for children")

TL;DR

This paper provides a detailed, visual, and archetypal approach to understanding the correspondence between closure operators and Lawvere-Tierney topologies on toposes, making complex proofs more intuitive and accessible.

Contribution

It introduces a visual and archetypal methodology for proving the correspondence between closure operators and topologies in toposes, including specific cases and visualization techniques.

Findings

Proofs on toposes with inclusions are simplified and visualized.

Visualization techniques help understand LT-topologies on specific toposes.

Proofs can be lifted from archetypal cases to general toposes.

Abstract

One of the main prerequisites for understanding sheaves on elementary toposes is the proof that a (Lawvere-Tierney) topology on a topos induces a closure operator on it, and vice-versa. That standard theorem is usually presented in a relatively brief way, with most details being left to the reader and with no hints on how to visualize some of the hardest axioms and proofs. These notes are, on a first level, an attempt to present that standard theorem in all details and in a visual way, following the conventions in "On my favorite conventions for drawing the missing diagrams in Category Theory" [Ochs2020]; in particular, some properties, like stability by pullbacks, are always drawn in the same "shape". On a second level these notes are also an experiment on doing these proofs on "archetypal cases" in ways that makes all the proofs easy to lift to the "general case". Our first archetypal case is a "topos with inclusions". This is a variant of the "toposes with canonical subobjects" from [Lambek/Scott 1986]; all toposes of the form $\mathbf{Set}^\mathbf{C}$, where $\mathbf{C}$ is a small category, are toposes with inclusions, and when we work with toposes with inclusions concepts like subsets and intersections are very easy to formalize. We do all our proofs on the correspondence between closure operators and topologies in toposes with inclusions, and then we show how to lift all our proofs to proofs that work on any topos. Our second archetypal case is toposes of the form $\mathbf{Set}^\mathbf{D}$, where $\mathbf{D}$ is a finite two-column graph. We show a way to visualize all the LT-topologies on toposes of the form $\mathbf{Set}^\mathbf{D}$, and we define formally a way to "add visual intuition to a proof about toposes"; this is related to the several techniques for doing "Category Theory for children" that are explained in "On my favorite conventions...".