Editing Monad (category theory) (section)

==== Algebras over the distribution monad ====
Another example is the '''''distribution monad''''' <math>\mathcal{D}</math> on the category of sets. It is defined by sending a set <math>X</math> to the set of functions <math>f : X \to [0,1]</math> with finite support and such that their sum is equal to <math>1</math>. In set-builder notation, this is the set<math display="block">\mathcal{D}(X) = \left\{
f: X \to [0,1] : \begin{matrix}
\#\text{supp}(f) < +\infty \\
\sum_{x \in X} f(x) = 1
\end{matrix}
\right\}</math>By inspection of the definitions, it can be shown that algebras over the distribution monad are equivalent to [[convex set]]s, i.e., sets equipped with operations <math>x +_r y</math> for <math>r \in [0,1]</math> subject to axioms resembling the behavior of convex linear combinations <math>rx + (1-r)y</math> in Euclidean space.<ref>{{Citation |last=Świrszcz |first=T. |title=Monadic functors and convexity|journal=Bull. Acad. Polon. Sci. Sér. Sci. Math. Astron. Phys.|volume=22 |year=1974 |pages=39–42|mr=0390019}},

{{Citation|doi=10.1007/978-3-642-15240-5_1|chapter=Convexity, Duality and Effects|title=Theoretical Computer Science|volume=323 |pages=1–19|series=IFIP Advances in Information and Communication Technology|year=2010|last1=Jacobs|first1=Bart |isbn=978-3-642-15239-9 |doi-access=free}}</ref>