Definition Convex Space [lcc-0020]

The category of convex spaces is the category of algebras for the monad \Delta : \mathsf {Set} \to \mathsf {Set} of discrete finite-support distributions. The morphisms are called \Delta -homomorphisms or homomorphisms of convex spaces.

So as not to multiply notation unnecessarily, we simply denote the category of convex spaces by \mathsf {Set}^\Delta , using the usual notation for the Eilenberg-Moore category.