# Spectral spaces versus distributive lattices: a dictionary

## TL;DR

This paper explores the classical anti-equivalence between distributive lattices and spectral spaces, providing a dictionary and methods to translate theorems into constructive forms even without clear pointwise content.

## Contribution

It offers a detailed dictionary for the anti-equivalence and demonstrates how to convert classical theorems into constructive versions using this correspondence.

## Key findings

- Examples illustrating the anti-equivalence
- A dictionary translating between the categories
- Constructive reformulations of classical theorems

## Abstract

The category of distributive lattices is, in classical mathematics, antiequivalent to the category of spectral spaces. We give here some examples and a short dictionary for this antiequivalence. We propose a translation of several abstract theorems (in classical mathematics) into constructive ones, even in the case where points of a spectral space have no clear constructive content. La cat\'egorie des treillis distributifs et celle des espaces spectraux sont anti\'equivalentes (en math\'ematiques classiques). Nous proposons ici un petit dictionnaire pour cette anti\'equivalence. Nous indiquons comment un certain nombre de th\'eor\`emes \'etranges des math\'ematiques classiques obtiennent un contenu constructif gr\^ace \`a cette anti\'equivalence, m\^eme dans le cas, fr\'equent, o\`u les points des espaces spectraux consid\'er\'es n'ont pas de contenu constructif clair.