Understand
We investigate the Eilenberg-Moore algebras of the extended probabilistic powerdomain monad $\mathcal V_w$ over the category $\mathbf{TOP}_0$ of $T_0$ topological spaces and continuous maps.
- We prove that every $\mathcal V_w$-algebra in our setting is a weakly locally convex sober topological cone, and that a map is the structure map of a $\mathcal V_w$-algebra if and only if it is continuous and sends every continuous valuation to its unique barycentre.
- Conversely, for locally linear sober cones (a strong form of local convexity), the mere existence of barycentres entails that the barycentre map is the structure map of a $\mathcal V_w$-algebra; moreover the algebra morphisms are exactly the linear continuous maps in that case.
- We also examine the algebras of two related monads, the simple valuation monad $\mathcal V_{\mathrm f}$ and the point-continuous valuation monad $\mathcal V_{\mathrm p}$.