In answer to Charles Wells
Can anyone tell me what Ehresmann meant by a "saturated functor" (foncteur saturé) in 1967?
Charles Ehresmann defined a "homomorphism saturated functor" already in his lectures in 1962, and it figures in his 1963 paper "Categories structurees" (Annales ENS), reprinted in "Charles Ehresmann: Oeuvres completes et Commentees", Part III-1, Amiens 1980, p. 29 In the "Comments" in this book I have given English translations in more modern terms of the main categorical definitions and results of Charles, which, up to the seventies, were often written in a non-usual style, very difficult to decipher to-day (and even at that moment for most readers, which explains they were not as widely known as they should have been!) In particular in the Note 29-2 (p. 348-9 of this book) I have translated the definition in more modern terms: A "homomorphism saturated functor" p: H -> C is a faithful amnestic functor which creates isomorphisms; amnestic means that an isomorphism mapped on an identity is an identity. Thus H is a concrete category over C, such that the restriction of p to the groupoid of isomorphisms of H is a discrete op-fibration. (there are more information in this Note).