Proposition [lcc-001L]

Let \Delta : \mathsf {Set} \to Set be the discrete probability distribution monad, so that its algebras are convex spaces. \Delta is finitary, so by abstract nonsense \mathsf {Alg}(\Delta ) has coproducts. These are given by 0 = \emptyset (with a trivial convex structure), and X + Y = X \sqcup X \times Y \times (0,1) \sqcup Y, with the convex structure given by regarding (x,y,\alpha ) as \alpha x + (1-\alpha ) y, and reducing any convex combination to one containing at most one element from each of X and Y.