Equilogical spaces and algebras for a double-power monad

Abstract

We investigate the algebras for the double-power monad on the Sierpisnki space in the category $\mathcal{Equ}$ of equilogical spaces, a cartesian closed extension of $\mathcal{Top}_0$ introduced by Scott, and the relationship of such algebras with frames. In particular, we focus our attention on interesting subcategories of $\mathcal{Equ}$. We prove uniqueness of the algebraic structure for a large class of equilogical spaces, and we characterize the algebras for the double-power monad in the category of algebraic lattices and in the category of continuous lattices, seen as full subcategories of $\mathcal{Equ}$.

We also analyse the case of algebras in the category $\mathcal{Top}_0$ of $\mathrm{T}_0$-spaces, again seen as a full subcategoy of $\mathcal{Equ}$, proving that each algebra for the double-power monad in $\mathcal{Top}_0$ has an underlying sober, compact, connected space.