On Eilenberg-Moore algebras induced by chains (Q5932486)

For a topological space \((S,T)\), let \(M_S\) be the induced monad on the category of sets, specified on objects by \[ M_S(A)=:\Hom_{\underline {{\mathcal T}op}}(S^A,S)=C(S^A). \] The problem of identifying explicitly the Eilenberg-Moore category of \(M_S\)-algebras has not been solved for a general topological space although many special cases have been treated: all classes when \(S\) is the Sierpiński dyad, the case of the unit interval with its usual topology, etc. The main result of the paper is the following representation theorem: Theorem 8.1. The Eilenberg-Moore category of \(M_S\)-algebras has, as objects, the \(C(S)\)-consistent dual frames and, as morphisms, the maps which preserve \(C(S)\)-action, arbitrary infima, finite suprema and constants. From the point of view of this representation, the mentioned particular cases are discussed.