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.
Categories: math.CO math.CT