Abstract
The continuous maps p : E → B for which p* : Top/B → Top/E reflects isomorphisms are shown to coincide with the universal quotient maps as characterized by Day and Kelly. Monadicity of p* turns out to be a local property. This is used to prove the main result of the paper, namely that p* is monadic for every locally sectionable map p : E → B. There are therefore important classes of maps p for which spaces over B are equivalently described as spaces over E which come equipped with a simple algebraic structure: local homeomorphisms, locally trivial quotient maps, surjective covering maps, etc. Finally, the monadic decomposition of p* is examined for arbitrary maps p.AMS Subject Classification18A3018C2018F1554C1055R10
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