##
**Displayed categories.**
*(English)*
Zbl 1419.18001

The article introduces the notion of display category. This concept is easily formalised as a category \(\mathcal{D}\) over \(\mathcal{C}\) together with a functor \(F: \mathcal{D} \to \mathcal{C}\) such that the objects of \(\mathcal{D}\) are given as a family indexed by \(\mathsf{Obj} \mathcal{C}\), and similarly for the morphisms, together with the obvious categorical constraints.

Then, the paper shows how creation of limits and Grothendieck fibrations, isofibrations, and discrete fibrations can be rendered by display categories. Finally, the article discusses how to use display categories within the univalent foundation framework.

At a first sight, the notion of display category appears to be very simple and unnecessary. However, the article convincingly shows that it should be understood as a natural, pragmatic tool, which encapsulates the core of most concrete examples of fibrations and their use. In this respect, the problematic issue of equality can be avoided, thus making the concept specifically useful in homotopy type theory, in which equality is a delicate notion requiring careful management. Moreover, the notion of display category is particularly well suited for computer formalisation and, in fact, all the results in the paper are also available as part of the \(\mathsf{UniMath}\) library for the \(\mathsf{Coq}\) proof assistant.

Summarising, this paper develops a natural and pragmatic tool which could be of interest to anyone working in univalent mathematics, and/or in the borderline between the formalisation and the semantics of type theories. The clean and sharp style of the article adds to the understanding and greatly helps whoever intends to apply the concept of display category in her own work.

Then, the paper shows how creation of limits and Grothendieck fibrations, isofibrations, and discrete fibrations can be rendered by display categories. Finally, the article discusses how to use display categories within the univalent foundation framework.

At a first sight, the notion of display category appears to be very simple and unnecessary. However, the article convincingly shows that it should be understood as a natural, pragmatic tool, which encapsulates the core of most concrete examples of fibrations and their use. In this respect, the problematic issue of equality can be avoided, thus making the concept specifically useful in homotopy type theory, in which equality is a delicate notion requiring careful management. Moreover, the notion of display category is particularly well suited for computer formalisation and, in fact, all the results in the paper are also available as part of the \(\mathsf{UniMath}\) library for the \(\mathsf{Coq}\) proof assistant.

Summarising, this paper develops a natural and pragmatic tool which could be of interest to anyone working in univalent mathematics, and/or in the borderline between the formalisation and the semantics of type theories. The clean and sharp style of the article adds to the understanding and greatly helps whoever intends to apply the concept of display category in her own work.

Reviewer: Marco Benini (Buccinasco)

### MSC:

18A15 | Foundations, relations to logic and deductive systems |

03B15 | Higher-order logic; type theory (MSC2010) |

### Keywords:

category theory; dependent type theory; computer proof assistants; Coq; univalent mathematics
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\textit{B. Ahrens} and \textit{P. Lefanu Lumsdaine}, Log. Methods Comput. Sci. 15, No. 1, Paper No. 20, 18 p. (2019; Zbl 1419.18001)

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### References:

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[7] | [CD13]Thierry Coquand and Nils Anders Danielsson, Isomorphism is equality, Indagationes Mathematicae 24 (2013), no. 4, 1105 – 1120, In memory of N.G. (Dick) de Bruijn (1918–2012), doi:10.1016/j. indag.2013.09.002. · Zbl 1359.03010 |

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