Grothendieck topology in nLab
2024年4月19日 · A Grothendieck topology may then be defined as a coverage that consists of sieves (which the Elephant calls “sifted”) and satisfies certain extra saturation conditions; see coverage for details.
A Grothendieck site is a category C together with a Grothendieck topology on C. Example 10. Let Xbe a topological space and let U be the collection of all open subsets of X, regarded as a partially ordered set with respect to inclusions. Then, when regarded as a category, the poset U carries a Grothendieck topology, where a collection of maps ...
4 Grothendieck topologies A Grothendieck site is a small category Cequipped with a topology T. A Grothendieck topology Tconsists of a collection of subfunctors Rˆhom( ;U); U2C; called covering sieves, such that the following ax-ioms hold: 1) (base change) If R ˆhom( ;U) is covering and ˚: V !Uis a morphism of C, then the subfunctor ˚ 1(R ...
When a category C is equipped with a Grothendieck topology we call it a site. These axioms don’t describe a topology using open sets, but in terms of covers. In the classical notion of a topology one needs to check that the given description of its open sets satisfies some properties with respect to intersections and unions. In
Toposes were originally introduced by Alexander Grothendieck in the early 1960s, in order to provide a mathematical underpinning for the ‘exotic’ cohomology theories needed in algebraic geometry. Every topological space gives rise to a topos and every topos in Grothendieck’s sense can be considered as a ‘generalized space’.
A Grothendieck topology Tconsists of the following data: a category, denoted CatT, along with a collection of covering sieves, denoted CovT. This means that, for each object
We will show that the classical proof system of geometric logic over a given geometric theory is equivalent to new proof systems based on the notion of Grothendieck topology.
The notion of Grothendieck topology: We can give two definitions - in terms of sieves, - in terms of covering families of morphisms. Definition. – (i) Let X be an object of a category C. A “sieve” over X is a collection C of morphisms to target X U u!X such that: For any element U u!X of C and any morphism V v!U of C,
We will first give a brief introduction on Grothendieck pretopologies, topologies and how they’re related. We will then generalize the concept of sheaves, normally used on topological spaces, to a category endowed with a (pre)topology and we’ll state an analogous result about the sheafification functor.
canonical topology in nLab
2023年4月16日 · The canonical topology on a category C C is the Grothendieck topology on C C which is the largest subcanonical topology. More explicitly, a sieve R R is a covering for the canonical topology iff every representable functor is a sheaf for every pullback of R R .
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