Grothendieck topology

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In category theory, a branch of mathematics, a Grothendieck topology is a structure on a category C {\displaystyle {\mathcal {C}}} that makes the objects of C {\displaystyle {\mathcal {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...

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