Fibrewise injectivity and Kock-Zöberlein monads
The paper deals with injectivity of continuous maps in topological \(T_0\)-spaces with respect to a fixed class of embeddings. The classes that are dealt with are the class of all embeddings, of all dense embeddings, of flat embeddings and of completely flat embeddings. Methods that have been developed by \textit{M. Escardo} and \textit{B. Flagg} [Electr. Notes Theor. Comput. Sci. 20, electronic paper No. 15 (1999; Zbl 0924.68116)] characterizing injectivity via Kock-Zöberlein monads and by \textit{D. Hofmann} [Order 30, No. 2, 643--655 (2013; Zbl 1282.54010)] applying these results in order to characterize injective objects with respect to these classes of embeddings in \(\mathsf{Top}_0,\) are lifted to the fiberwise setting. The method involves the use of different filter monads \(\mathbb{T}\) and associated \(\mathbb{T}\) way below relations. The filter monads are carried into the comma category \(\mathsf{Top}_0 \downarrow Z\) of \(T_0\)-topological spaces over \(Z,\) so that they inherit the Kock-Zöberlein property and allow for fibrewise \(\mathbb{T}\) way below relations. The main result characterizes continuous maps injective with respect to \(\mathbb{T}\) embeddings as fibrewise sober, fibrewise \(\mathbb{T}\) core compact and fibrewise \(\mathbb{T}\) stable maps. Applications to injectivity of continuous maps in \(\mathsf{Top}\) with respect to the same classes of embeddings are deduced from it. Moreover it is shown that together with the corresponding embeddings, injective continuous maps form a weak factorization system.
scientific article; zbMATH DE number 6138021
scientific article