On a formula that is not in "Grothendieck Topologies in Posets"

TL;DR

This paper clarifies the explicit formula for converting a nucleus to a subset within the context of Grothendieck topologies on Artinian posets, filling a gap in the existing theoretical framework.

Contribution

It provides the explicit formula for the nucleus-to-subset conversion, enhancing understanding of the bijections in Grothendieck topologies on Artinian posets.

Findings

Explicit formula for nucleus to subset conversion

Completes the bijection framework in the existing theory

Clarifies the relationship between nuclei and subsets

Abstract

The paper "Grothendieck Topologies on Posets" by A.J. Lindenhovius shows that when $\mathbf{P}$ is an Artinian poset and $\mathbf{E}$ is the topos $\mathbf{Set}^\mathbf{P}$ then there are bijections between the set of subsets of $\mathbf{P}$, the set of Grothendieck topologies on $\mathbf{E}$, and the set of nuclei on the Heyting Algebra $\mathrm{Sub}(1_\mathbf{E})$. It also shows that there are nice formulas for converting between subsets, Grothendieck topologies, and nuclei, but the formula for converting a nucleus to a subset is not spelled out explicitly. These notes fix that gap.