Grothendieck originally introduced the machinery of Grothendieck topologies and topoi to define the étale topology. In this language, the definition of the étale topology is succinct but abstract: It is the topology generated by the pretopology whose covering families are jointly surjective families of étale morphisms. The small étale site of X is the category "O"("X"ét) whose objects are schemes "U" with a fixed étale morphism "U" → "X". The morphisms are morphisms of schemes compatible with the fixed maps to "X". The big étale site of X is the category "Ét/X", that is, the category of schemes with a fixed map to "X", considered with the étale topology.