# Grothendieck topologies on posets

@article{Hemelaer2018GrothendieckTO, title={Grothendieck topologies on posets}, author={Jens Hemelaer}, journal={arXiv: Category Theory}, year={2018} }

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…

## 3 Citations

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### Filters and compactness on small categories and locales

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In analogy with the classical theory of filters, for finitely complete or small categories, we provide the concepts of filter, \(\mathfrak{G}\)-neighborhood (short for "Grothendieck-neighborhood")…

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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…

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