# 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

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

- Mathematics
- 2021

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…

### Filters and compactness on small categories and locales

- MathematicsOpen Journal of Mathematical Sciences
- 2022

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")…

### Each closure operator induces a topology and vice-versa ("version for children")

- Mathematics
- 2021

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

### Grothendieck topologies on a poset

- Mathematics
- 2014

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…

### Points in the fppf topology

- Mathematics
- 2014

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…

### Azumaya toposes

- Mathematics
- 2017

In [4], many different Grothendieck topologies were introduced on the category of Azumaya algebras. Here we give a classification in terms of sets of supernatural numbers. Then we discuss the…

### Azumaya representation schemes

- 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.

### A topos-theoretic approach to Stone-type dualities

- Mathematics
- 2011

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…

### Prime ideal structure in commutative rings

- Mathematics
- 1969

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…

### Continuous Lattices and Domains

- 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…

### Non-Hausdorff topology and domain theory

- Mathematics
- 2013

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…