• Corpus ID: 119146922

Grothendieck topologies on posets

  title={Grothendieck topologies on posets},
  author={Jens Hemelaer},
  journal={arXiv: Category Theory},
  • 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… 

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

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

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")

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



Grothendieck topologies on a poset

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

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

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

Points in algebraic geometry

Azumaya representation schemes

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

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

D-completions and the d-topology

Prime ideal structure in commutative rings

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

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

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