On exponentiability of étale algebraic homomorphisms

• Autor(a) principal: Clementino, Maria Manuel
• Data de Publicação: 2013
• Outros Autores: Hofmann, Dirk, Janelidze, George
• Tipo de documento: Artigo
• Idioma: eng
• Título da fonte: Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)
• Texto Completo: http://hdl.handle.net/10316/43640; https://doi.org/10.1016/j.jpaa.2012.10.013

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 Alg(T) of Eilenberg-Moore T-algebras, for a monad T, and proved in case T satisfies the so-called Beck-Chevalley condition. For that, Alg(T) is embedded in the (topological) category RelAlg(T) 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 RelAlg(T), and, moreover, these exponentials belong to Alg(T) whenever the morphisms are étale.