  In this article, we characterize convexity in terms of algebras over a PROP,
and establish a tensor-product-like symmetric monoidal structure on the
category of convex sets. Using these two structures, and the theory of
$\scr{O}$-monoidal categories, we state and prove a Grothendieck construction
for lax $\scr{O}$-monoidal functors into convex sets. We apply this
construction to the categorical characterization of entropy of Baez, Fritz, and
Leinster, and to the study of quantum contextuality in the framework of
simplicial distributions.
