It is a Cauchy-complete geometric site. We prove that limits and colimits exist in k - Space . Y are morphisms of k -spaces. Limits and colimits exist in the category of k-spaces. It remains to prove that fibre coproducts exist in k - Space .
Y are two morphisms of k -spaces.
The fibre product of separated spaces is separated. We now use the notion of k -spaces to define an algebraic variety in the sense of Serre. First we define affine spaces over k . An affine variety is a k -space which is isomorphic to a closed subspace of an affine space over k . U i is an affine k -variety.