As far as we know, no notion of concrete fibration is available. We provide one such notion in adherence to the foundational attitude that characterizes the adoption of the fibrational perspective in approaching fundamental subjects in category theory and discuss it in connection with the notion of concrete category and the notions of locally small and small fibrations. We also discuss the appropriateness of our notion of concrete fibration for fibrations of small maps, which is relevant to algebraic set theory.
"Concrete Fibrations." Notre Dame J. Formal Logic 58 (2) 179 - 204, 2017. https://doi.org/10.1215/00294527-3817788