• Corpus ID: 119146922

# Grothendieck topologies on posets

```@article{Hemelaer2018GrothendieckTO,
title={Grothendieck topologies on posets},
author={Jens Hemelaer},
journal={arXiv: Category Theory},
year={2018}
}```
• Jens Hemelaer
• Published 25 November 2018
• Mathematics
• arXiv: Category Theory
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
The paper [Lin14] shows that when P is an Artinian poset and E is the topos Set then there are bijections between the set of subsets of P, the set of Grothendieck topologies on E, and the set of
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")
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

## References

SHOWING 1-10 OF 21 REFERENCES

We investigate Grothendieck topologies (in the sense of sheaf theory) on a poset \$\P\$ that are generated by some subset of \$\P\$. We show that such Grothendieck topologies exhaust all possibilities if
Using methods from commutative algebra and topos-theory, we construct topos-theoretical points for the fppf topology of a scheme. These points are indexed by both a geometric point and a limit
• Mathematics
• 2016
It is shown that the functor assigning to an Azumaya algebra \$A\$ the set of all algebra maps from a fixed \$\mathbb{C}\$-algebra \$R\$, is a sheaf for all such Grothendieck topologies coarser than the maximal flat topology.
We present an abstract unifying framework for interpreting Stone-type dualities; several known dualities are seen to be instances of just one topos-theoretic phenomenon, and new dualities are
0. Introduction. Let ' be the category of commutative rings with unit, and regard Spec (as in [1]) as a contravariant functor from ' to g\$7 the category of topological spaces and continuous maps. The
• Mathematics
• 2003
Preface Acknowledgements Foreword Introduction 1. A primer on ordered sets and lattices 2. Order theory of domains 3. The Scott topology 4. The Lawson Topology 5. Morphisms and functors 6. Spectral
1. Introduction 2. Elements of set theory 3. A first tour of topology: metric spaces 4. Topology 5. Approximation, and function spaces 6. Metrics, quasi-metrics, hemi-metrics 7. Completeness 8. Sober
• Computer Science
Cambridge tracts in theoretical computer science
• 1998
This chapter discusses the development of lambda-calculi in CCC's of algebraic dcpo's, as well as its applications in recursion theory and category theory.