Abstract
We prove a universal property for ∞-categories of spans in the generality of Barwick’s adequate triples, explicitly describe the cocartesian fibration corresponding to the span functor, and show that the latter restricts to a self-equivalence on the class of orthogonal adequate triples, which we introduce for this purpose.
As applications of the machinery we develop, we give a quick proof of Barwick’s unfurling theorem, show that an orthogonal factorisation system arises from a cartesian fibration if and only if it forms an adequate triple (generalising work of Lanari), extend the description of dual (co)cartesian fibrations by Barwick, Glasman and Nardin to two-variable fibrations, explicitly describe parametrised adjoints (extending work of Torii), identify the orthofibration classifying the mapping category functor of an (∞,2)-category (building on work of Abellán García and Stern), formally identify the unstraightenings of the identity functor on the ∞-category of ∞-categories with the (op)lax under-categories of a point, and deduce a certain naturality property of the Yoneda embedding (answering a question of Clausen).
As applications of the machinery we develop, we give a quick proof of Barwick’s unfurling theorem, show that an orthogonal factorisation system arises from a cartesian fibration if and only if it forms an adequate triple (generalising work of Lanari), extend the description of dual (co)cartesian fibrations by Barwick, Glasman and Nardin to two-variable fibrations, explicitly describe parametrised adjoints (extending work of Torii), identify the orthofibration classifying the mapping category functor of an (∞,2)-category (building on work of Abellán García and Stern), formally identify the unstraightenings of the identity functor on the ∞-category of ∞-categories with the (op)lax under-categories of a point, and deduce a certain naturality property of the Yoneda embedding (answering a question of Clausen).
| Original language | English |
|---|---|
| Article number | e111 |
| Pages (from-to) | 1-70 |
| Number of pages | 70 |
| Journal | Forum of Mathematics, Sigma |
| Volume | 11 |
| Issue number | 2023 |
| Early online date | 7 Dec 2023 |
| DOIs | |
| Publication status | Published - 7 Dec 2023 |
Bibliographical note
Open Access via the CUP agreementFunding statement
During the preparation of this text, FH and SL were members of the Hausdorff Center for Mathematics at the University of Bonn funded by the German Research Foundation (DFG), and FH was furthermore a member of the cluster ‘Mathematics Münster: Dynamics-Geometry-Structure’ at the University of Münster under grant nos. EXC 2047 and EXC 2044, respectively. FH would also like to thank the Mittag-Leffler Institute for its hospitality during the research program ‘Higher Algebraic Structures in Algebra, Topology and Geometry’, supported by the Swedish Research Council (VR) under grant no. 2016-06596. FH and JN were further supported by the European Research Council (ERC) through the grants ‘Moduli Spaces, Manifolds and Arithmetic’, grant no. 682922, and ‘Derived Symplectic Geometry and Applications’, grant no. 768679, respectively. SL was supported by the DFG Schwerpunktprogramm 1786 ‘Homotopy Theory and Algebraic Geometry’ (project ID SCHW 860/1-1).