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**Concrete fibrations.**
*(English)*
Zbl 1386.18001

The paper analyses the notion of concrete fibration. The significance of the work stems fron the fact that such notion has never been studied before despite it is strictly connected with the concepts of concrete category and locally small and small fibrations.

The key point hinges on the characterisation of concreteness via the so-called Isbell condition: For every pair of objects, the class of spans between the objects modulo a suitable equivalence relation admits a system of representatives which is a proper set, so that each span in the class is equivalent to exactly one span in the set.

The paper, after a well argumented introduction, briefly analyses concrete categories, leading to the notorius result: A category is concretizable if and only if it satisfies the Isbell condition. Then, the basics of fibered categories in the spirit of Bénabou are recalled.

The Isbell condition is then reformulated for fibrations, and specialised for cloven fibrations as a first-order categorial universal property. After that, under the definability of isomorphisms, a fibration satisfying the Isbell condition is shown to be locally small. This means that definability is roughly the fibrational counterpart of the axiom of separation in set theory.

To illustrate the significance of the obtained results and to strongly motivate them, concrete fibrations of small maps and their sub-fibrations are studied, which has direct applications in algebraic set theory.

In the overall, the article is quite technical, although the author makes a successful effort to support the results with a good intuitive guide. Relevant for categorists and experts in algebraic set theory, it also provides a concise and clear introduction to the fibrational approach and its spirit, which could be of interest for the general reader.

The key point hinges on the characterisation of concreteness via the so-called Isbell condition: For every pair of objects, the class of spans between the objects modulo a suitable equivalence relation admits a system of representatives which is a proper set, so that each span in the class is equivalent to exactly one span in the set.

The paper, after a well argumented introduction, briefly analyses concrete categories, leading to the notorius result: A category is concretizable if and only if it satisfies the Isbell condition. Then, the basics of fibered categories in the spirit of Bénabou are recalled.

The Isbell condition is then reformulated for fibrations, and specialised for cloven fibrations as a first-order categorial universal property. After that, under the definability of isomorphisms, a fibration satisfying the Isbell condition is shown to be locally small. This means that definability is roughly the fibrational counterpart of the axiom of separation in set theory.

To illustrate the significance of the obtained results and to strongly motivate them, concrete fibrations of small maps and their sub-fibrations are studied, which has direct applications in algebraic set theory.

In the overall, the article is quite technical, although the author makes a successful effort to support the results with a good intuitive guide. Relevant for categorists and experts in algebraic set theory, it also provides a concise and clear introduction to the fibrational approach and its spirit, which could be of interest for the general reader.

Reviewer: Marco Benini (Buccinasco)