Caviglia, Elena; Mesiti, Luca Indexed Grothendieck construction. (English) Zbl 07898687 Theory Appl. Categ. 41, 894-926 (2024). Summary: We produce an indexed version of the Grothendieck construction. This gives an equivalence of categories between opfibrations over a fixed base in the 2-category of 2-copresheaves and 2-copresheaves on the Grothendieck construction of the fixed base. We also prove that this equivalence is pseudonatural in the base and that it restricts to discrete opfibrations with small fibres and copresheaves. Our result is a 2-dimensional generalization of the equivalence between slices of copresheaves and copresheaves on slices. We can think of the indexed Grothendieck construction as a simultaneous Grothendieck construction on every index that takes into account all bonds between different indexes. MSC: 18D30 Fibered categories 18N10 2-categories, bicategories, double categories 18F20 Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects) 18B25 Topoi 18A30 Limits and colimits (products, sums, directed limits, pushouts, fiber products, equalizers, kernels, ends and coends, etc.) Keywords:Grothendieck construction; indexed; category of elements; fibration; presheaf; 2-category × Cite Format Result Cite Review PDF Full Text: arXiv Link References: [1] B. Ahrens, P. R. North, and N. van der Weide. Bicategorical type theory: semantics and syntax. Mathematical Structures in Computer Science, 33(10):868-912, 2023. · Zbl 07813371 [2] S. Awodey. On Hofmann-Streicher universes. arXiv: https: // arxiv. org/ abs/ 2205. 10917 , 2023. [3] F. Borceux. Handbook of Categorical Algebra, volume 2 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, 1994. · Zbl 0803.18001 [4] S. E. Crans. A tensor product for Gray-categories. Theory Appl. Categ., 5:12-69, 1999. · Zbl 0914.18006 [5] A. Grothendieck and M. Raynaud. Revêtements Etales et Groupe Fondamental -Séminaire de Géométrie Algébrique du Bois Marie 1960/61 (SGA 1). Lect. Notes Math. Springer Berlin Heidelberg, 1971. · Zbl 0234.14002 [6] M. Hofmann and T. Streicher. Lifting Grothendieck universes. preprint: https: // www2. mathematik. tu-darmstadt. de/ streicher/ NOTES/ lift. pdf , 1997. [7] B. Jacobs. Categorical Logic and Type Theory, volume 141 of Stud. Logic Found. Math. Amster-dam: Elsevier, 1999. · Zbl 0911.03001 [8] G. M. Kelly. Basic concepts of enriched category theory. Repr. Theory Appl. Categ., 2005(10):1-136, 2005. · Zbl 1086.18001 [9] L. Mesiti. Pointwise Kan extensions along 2-fibrations and the 2-category of elements. arXiv: https: // arxiv. org/ abs/ 2302. 04566 , 2023. [10] L. Mesiti. 2-classifiers via dense generators and Hofmann-Streicher universe in stacks. arXiv: https: // arxiv. org/ abs/ 2401. 16900 , 2024. [11] R. Street. Fibrations and Yoneda’s Lemma in a 2-category. Category Sem., Proc., Sydney 1972/1973, Lect. Notes Math. 420, 104-133, Springer, 1974. · Zbl 0327.18006 [12] R. Street. Limits indexed by category-valued 2-functors. J. Pure Appl. Algebra, 8:149-181, 1976. · Zbl 0335.18005 [13] M. Weber. Yoneda structures from 2-toposes. Appl. Categ. Struct., 15(3):259-323, 2007. · Zbl 1125.18001 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.