Site (mathematics) | Wikipedia audio article
Channel: wikipedia tts
Date: 07/10/2019

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Grothendieck_topology


00:00:31 1 Overview
00:01:03 2 Definition
00:01:19 2.1 Motivation
00:02:07 2.2 Sieves
00:02:39 2.3 Grothendieck topology
00:03:10 2.3.1 Axioms
00:03:42 2.3.2 Grothendieck pretopologies
00:04:14 3 Sites and sheaves
00:04:46 4 Examples of sites
00:05:02 4.1 The discrete and indiscrete topologies
00:05:34 4.2 The canonical topology
00:06:05 4.3 Small site associated to a topological space
00:06:37 4.4 Big site associated to a topological space
00:07:09 4.5 The big and small sites of a manifold
00:07:41 4.6 Topologies on the category of schemes
00:08:13 5 Continuous and cocontinuous functors
00:08:45 5.1 Continuous functors
00:09:16 5.2 Cocontinuous functors
00:09:48 5.3 Morphisms of sites
00:10:36 6 See also
00:11:08 7 Notes



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SUMMARY
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In category theory, a branch of mathematics, a Grothendieck topology is a structure on a category C which makes the objects of C act like the open sets of a topological space.  A category together with a choice of Grothendieck topology is called a site.
Grothendieck topologies axiomatize the notion of an open cover.  Using the notion of covering provided by a Grothendieck topology, it becomes possible to define sheaves on a category and their cohomology.  This was first done in algebraic geometry and algebraic number theory by Alexander Grothendieck to define the étale cohomology of a scheme.  It has been used to define other cohomology theories since then, such as l-adic cohomology, flat cohomology, and crystalline cohomology. While Grothendieck topologies are most often used to define cohomology theories, they have found other applications as well, such as to John Tate's theory of rigid analytic geometry.
There is a natural way to associate a site to an ordinary topological space, and Grothendieck's theory is loosely regarded as a generalization of classical topology.  Under meager point-set hypotheses, namely sobriety, this is completely accurate—it is possible to recover a sober space from its associated site.  However simple examples such as the indiscrete topological space show that not all topological spaces can be expressed using Grothendieck topologies.  Conversely, there are Grothendieck topologies which do not come from topological spaces.
The term "Grothendieck topology" has changed in meaning. In Artin (1962) it meant what is now called a Grothendieck pretopology, and some authors still use this old meaning. Giraud (1964) modified the definition to use sieves rather than covers. Much of the time this does not make much difference, as each Grothendieck pretopology determines a unique Grothendieck topology, though quite different pretopologies can give the same topology.