Grothendieck topologies on posets

TL;DR

This paper extends the classification of Grothendieck topologies on posets beyond Artinian cases, providing explicit descriptions and cardinalities for various families, by translating locale and domain theory results.

Contribution

It generalizes Lindenhovius's classification to broader posets using locale and domain theory, offering explicit descriptions and cardinality computations.

Findings

Explicit descriptions of Grothendieck topologies with enough points

Cardinality calculations for various poset examples

Extension of classification beyond Artinian posets

Abstract

Lindenhovius has studied Grothendieck topologies on posets and has given a complete classification in the case that the poset is Artinian. We extend his approach to more general posets, by translating known results in locale and domain theory to the study of Grothendieck topologies. In particular, explicit descriptions are given for the family of Grothendieck topologies with enough points and the family of Grothendieck topologies of finite type. As an application, we compute the cardinalities of these families in various examples.