Distributive law is a way to compose two algebraic structures, say $𝒰$ and $𝒱$, into a more complex algebraic structure $𝒲$. The aim of this paper is to understand distributive laws in terms of operads. The central result says that if the operads corresponding respectively to $𝒰$ and $𝒱$ are Koszul, then the operad corresponding to $𝒲$ is Koszul as well. An application to the cohomology of configuration spaces is given.

The paper establishes a duality between a category of free subreducts of affine spaces and a corresponding category of generalized hypercubes with constants. This duality yields many others, in particular a duality between the category of (finitely generated) free barycentric algebras (simplices of real affine spaces) and a corresponding category of hypercubes with constants.