A version of Loday's "coquecigrue" problem over arbitrary ground fields seeks analogues of affine algebraic groups whose tangent spaces are Leibniz algebras. To that end, we construct functors assigning left and right Leibniz algebras to pointed rack objects in the category of affine schemes. These functors have many desirable properties; in particular, they recover the Lie algebras of linear algebraic groups (via conjugation quandles) and the Leibniz algebras of algebraic Lie racks. We also use rack schemes to functorially construct (co-)nondegenerate solutions to the Yang–Baxter equation in the categories of schemes, sets, and commutative k-algebras.