On exponentiability of étale algebraic homomorphisms

In this paper we show that the theorem, by Cagliari and Mantovani, stating that in the category of compact Hausdorff spaces every étale map is exponentiable, can be formulated in a general category of Eilenberg–Moore T-algebras, for a monad T, and proved in case T satisfies the so-called Beck–Chevalley condition. For that, is embedded in the (topological) category of relational T-algebras, where a suitable notion of étale morphism can be studied, it is shown that morphisms between T-algebras are exponentiable in , and, moreover, these exponentials belong to whenever the morphisms are étale.