Fibrewise injectivity and Kock-Zöberlein monads
type: Item
instance of: scholarly article
title: Fibrewise injectivity and Kock-Zöberlein monads
zbMATH Open document ID: 1280.54010
author: Maria Manuel Clementino
author: Francesca Cagliari
author: Sandra Mantovani
published in: Journal of Pure and Applied Algebra
publication date: 2013-02-21T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 54C10
Mathematics Subject Classification ID: 54B30
Mathematics Subject Classification ID: 54A20
Mathematics Subject Classification ID: 18C15
Mathematics Subject Classification ID: 54F65
Mathematics Subject Classification ID: 06B35
zbMATH DE Number: 6138021
zbMATH Keywords: injective continuous map
zbMATH Keywords: filter monad
zbMATH Keywords: Kock-Zöberlein monad
zbMATH Keywords: comma category
zbMATH Keywords: way below relation
zbMATH Keywords: fibre wise injectivity
zbMATH Keywords: weak factorization system
reviewed by: Eva Colebunders
MaRDI profile type: Publication
full work available at URL: j.jpaa.2012.03.019
OpenAlex ID: W2014413072
Wikidata QID: Q122758467
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DOI: 10.1016/J.JPAA.2012.03.019
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description: scientific article; zbMATH DE number 6138021